CHAPTER 1 PSEUDO-DIFFERENTIAL OPERATORS Introduction In the first chapter, we develop the analysis needed to define the index of an elliptic operator and to compute the index using heat equation meth- ods. Sections 1.1 and 1.2 are brief reviews of Sobolev spaces and pseudo- differential operators on Euclidean spaces. In section 1.3, we transfer these notions to compact Riemannian manifolds using partition of unity argu- ments. In section 1.4 we review the facts concerning Fredholm operators needed in section 1.5 to prove the Hodge decomposition theorem and to discuss the spectral theory of self-adjoint elliptic operators. In section 1.6 we introduce the heat equation and in section 1.7 we derive the local for- mula for the index of an elliptic operator using heat equation methods. Section 1.8 generalizes the results of section 1.7 to find a local formula for the Lefschetz number of an elliptic complex. In section 1.9, we dis- cuss the index of an elliptic operator on a manifold with boundary and in section 1.10, we discuss the zeta and eta invariants. Sections 1.1 and 1.4 review basic facts we need, whereas sections 1.8 through 1.10 treat advanced topics which may be omitted from a first reading. We have attempted to keep this chapter self-contained and to assume nothing beyond a first course in analysis. An exception is the de Rham theorem in section 1.5 which is used as an example. A number of people have contributed to the mathematical ideas which are contained in the first chapter. We were introduced to the analysis of sections 1.1 through 1.7 by a course taught by L. Nirenberg. Much of the organization in these sections is modeled on his course. The idea of using the heat equation or the zeta function to compute the index of an elliptic operator seems to be due to R. Bott. The functional calculus used in the study of the heat equation contained in section 1.7 is due to R. Seeley as are the analytic facts on the zeta and eta functions of section 1.10. The approach to Lefschetz fixed point theorems contained in section 1.8 is due to T. Kotake for the case of isolated fixed points and to S. C. Lee and the author in the general case. The analytic facts for boundary value problems discussed in section 1.9 are due to P. Greiner and R. Seeley.

1.1. Fourier Transform, Schwartz Class, And Sobolev Spaces. The Sobolev spaces and Fourier transform provide the basic tools we shall need in our study of elliptic partial differential operators. Let x = (x_{1},…, x_{m}) ∈ R^{m}. If x, y ∈ R^{m}, we define: x · y = x^{1}y^{1} + ⋯ + x^{m}y^{m}  and |x| = (x · x)1/2 as the Euclicean dot product and length. Let α = (α_{1},…, α_{m}) be a multi- index. The α_{j} are non-negative integers. We define: |α| = α_{1} + ⋯ + α_{m} ,  α! = α_{1}! … α_{m}! ,  x^{α} = x_{α1 1} … x_{αm m}. Finally, we define: d_{α x} = ( ∂ ∂x1 )^{α1}… ( ∂ ∂xm )^{αm}  and D_{α x} = (−i)^{|α|}d^{α} x as a convenient notation for multiple partial differentiation. The extra factors of (−i) defining D_{α x} are present to simplify later formulas. If f(x) is a smooth complex valued function, then Taylor’s theorem takes the form: f(x) = _{∑ |α|≤n} d_{α x}f(x_{0})(x − x^{0})α α! + O(|x − x_{0}|^{n+1}). The Schwartz class S is the set of all smooth complex valued functions f on R^{m} such that for all α, β there are constants C_{α,β} such that |x^{α}D_{β x}f| ≤ C_{α,β}. This is equivalent to assuming there exist estimates of the form: |D^{β x}f| ≤ C^{n,β}(1 + |x|)−n for all (n, β). The functions in S have all their derivatives decreasing faster at ∞ than the inverse of any polynomial. For the remainder of Chapter 1, we let dx, dy, dξ, etc., denote Lebesgue measure on R^{m} with an additional normalizing factor of (2π)^{−m/2}. With this normalization, the integral of the Gaussian distribution becomes: ∫ e^{−_{^{1} 2} |x|2} dx = 1. We absorb the normalizing constant into the measure in order to sim- plify the formulas of the Fourier transform. If C_{∞ 0}(R^{m}) denotes the set of smooth functions of compact support on R^{m}, then this is a subset of S. Since C_{0}(R^{m}) is dense in L^{2}(R^{m}), S is dense in L^{2}(R^{m}). We define the convolution product of two elements of S by: (f ∗ g)(x) = ∫ f(x − y)g(y) dy = ∫ f(y)g(x − y) dy. This defines an associative and commutative multiplication. Although there is no identity, there do exist approximate identities:

Lemma 1.1.1. Let f ∈ S with ∫ f(x) dx = 1. Define f_{u}(x) = u^{−m}f(_{x u}). Then for any g ∈ S, f_{u} ∗ g converges uniformly to g as u → 0. Proof: Choose C so ∫ |f(x)| dx ≤ C and |g(x)| ≤ C. Because the first derivatives of g are uniformly bounded, g is uniformly continuous. Let ε > 0 and choose δ > 0 so |x − y| ≤ δ implies |g(x) − g(y)| ≤ ε. Because ∫ f_{u}(x) dx = 1, we compute: |f_{u} ∗ g(x) − g(x)| = ||||∫ f_{u}(y){g(x − y) − g(x)} dy|||| ≤ ∫ |f_{u}(y){g(x − y) − g(x)}| dy. We decompose this integral into two pieces. If |y| ≤ δ we bound it by Cε. The integral for |y| ≥ δ can be bounded by: 2C ∫_{|y|≥δ} |f_{u}(y)| dy = 2C ∫_{|y|≥δ/u} |f(y)| dy. This converges to zero as u → 0 so we can bound this by Cε if u < u(ε). This completes the proof. A similar convolution smoothing can be applied to approximate any ele- ment of L^{p} arbitrarily well in the L^{p} norm by a smooth function of compact support. We define the Fourier transform f̂(ξ) by: f̂(ξ) = ∫ e^{−ix·ξ} f(x) dx for f ∈ S. For the moment ξ ∈ R^{m}; when we consider operators on manifolds, it will be natural to regard ξ as an element of the fiber of the cotangent space. By integrating by parts and using Lebesgue dominated convergence, we compute:D_{α ξ}{f̂(ξ)} = (−1)^{|α|}{x^{α}f} and ξ^{α}f̂(ξ) = {D_{α x}f}. This implies f̂ ∈ S so Fourier transform defines a map S → S. We compute the Fourier transform of the Gaussian distribution. Let f_{0}(x) = exp(−_{1 2} |x|^{2}), then f_{0} ∈ S and ∫ f_{0}(x) dx = 1. We compute: f̂0(ξ) = ∫ e^{−ix·ξ} e^{−_{^{1} 2} |x|2} dx = e^{−_{^{1} 2} |ξ|2} ∫ e^{−(x+iξ)·(x+iξ)/2} dx.

4 1.1. Fourier Transform, Schwartz Class We make a change of variables to replace x + iξ by x and to shift the contour in C^{m} back to the original contour R^{m}. This shows the integral is 1 and f̂0(ξ) = exp (−_{1 2} |ξ|^{2}) so the function f_{0} is its own Fourier transform. In fact, the Fourier transform is bijective and the Fourier inversion for- mula gives the inverse expressing f in terms of f̂ by: f(x) = ∫ e^{ix·ξ} f̂(ξ) dξ = ˆ f̂(−x). We define T (f) = ˆ f̂(−x) = ∫ e^{ix·ξ} f̂(ξ) dξ as a linear map from S → S. We must show that T (f) = f to prove the Fourier inversion formula. Suppose first f(0) = 0. We expand: f(x) = ∫_{1 0} d dt{f(tx)} dt = _{∑ j} x_{j} ∫_{1 0} ∂f ∂xj (tx) dt = _{∑ j} x_{j}gj where the g_{j} are smooth. Let ϕ ∈ C_{∞ 0}(R^{m}) be identically 1 near x = 0. Then we decompose: f(x) = ϕf(x) + (1 − ϕ)f(x) = _{∑ j} x_{j}ϕg_{j} +_{∑ j} x_{j} {x_{j}(1 − ϕ)f |x|2 } . Since ϕg_{j} has compact support, it is in S. Since ϕ is identically 1 near x = 0, x_{j}(1 − ϕ)f/|x|^{2} ∈ S. Thus we can decompose f = ∑ x_{j}h_{j} for h_{j} ∈ S. We Fourier transform this identity to conclude: f̂ = _{∑ j} {x_{j}h_{j}} = _{∑ j} i∂ĥj ∂ξj . Since this is in divergence form, T (f)(0) = ∫ f̂(ξ) dξ = 0 = f(0). More generally, let f ∈ S be arbitrary. We decompose f = f(0)f_{0} + (f − f(0)f_{0}) for f_{0} = exp(−_{1 2} |x|^{2}). Since f̂0 = f_{0} is an even function, T (f_{0}) = f_{0} so that T (f)(0) = f(0)f_{0}(0)+T (f −f(0)f_{0}) = f(0)f_{0}(0) = f(0) since (f − f(0)f_{0})(0) = 0. This shows T (f)(0) = f(0) in general. We use the linear structure on R^{m} to complete the proof of the Fourier inversion formula. Let x_{0} ∈ R^{m} be fixed. We let g(x) = f(x + x_{0}) then: f(x_{0}) = g(0) = T (g)(0) = ∫ e^{−ix·ξ} f(x + x_{0}) dx dξ = ∫ e^{−ix·ξ} e^{ix_{0}·ξ} f(x) dx dξ = T (f)(x_{0}).

And Sobolev Spaces 5

This shows the Fourier transform defines a bijective map S → S. If we use the constants C_{α,β} = sup_{x∈Rm} |x^{α}D_{β x}f| to define a Frechet structure on S, then the Fourier transform is a homeomorphism of topological vector spaces. It is not difficult to show C_{∞ 0}(R^{m}) is a dense subset of S in this topology. We can use either pointwise multiplication or convolution to define a multiplication on S and make S into a ring. The Fourier transform interchanges these two ring structures. We compute: f̂ · ĝ = ∫ e^{−ix·ξ} f(x)e^{−iy·ξ} g(y) dx dy = ∫ e^{−i(x−y)·ξ} f(x − y)e^{−iy·ξ} g(y) dx dy = ∫ e^{−ix·ξ} f(x − y)g(y) dx dy. The integral is absolutely convergent so we may interchange the order of integration to compute f̂ · ĝ = (f ∗ g). If we replace f by f̂ and g by ĝ we see (f · g)(−x) = (f̂ ∗ ĝ) using the Fourier inversion formula. We now take the Fourier transform and use the Fourier inversion formula to see (f · g)(−ξ) = (f̂ ∗ ĝ)(−ξ) so that (f · g) = f̂ ∗ ĝ. The final property we shall need of the Fourier transform is related to the L^{2} inner product (f, g) = ∫ f(x)g(x) dx. We compute: (f̂, g) = ∫ f(x)e^{−ix·ξ} g(ξ) dx dξ = ∫ f(x)e^{−ix·ξ} g(ξ) dξ dx = (f, ĝ(−x)). If we replace g by ĝ then (f̂, ĝ) = (f, ˆ ĝ(−x)) = (f, g) so the Fourier trans- form is an isometry with respect to the L^{2} inner product. Since S is dense in L^{2}, it extends to a unitary map L^{2}(R^{m}) → L^{2}(R^{m}). We summarize these properties of the Fourier transform as follows: Lemma 1.1.2. The Fourier transform is a homeomorphism S → S such that: (a) f(x) = ∫ e^{ix·ξ} f̂(ξ) dξ = ∫ e^{i(x−y)·ξ} f(y) dy dξ (Fourier inversion for- mula); (b) D_{α x}f(x) = ∫ e^{ix·ξ} ξ^{α}f̂(ξ) dξ and ξ^{α}f̂(ξ) = ∫ e^{−ix·ξ} D_{α x}f(x) dx; (c) f̂ · ĝ = (f ∗ g) and f̂ ∗ ĝ = (f · g); (d) The Fourier transform extends to a unitary map of L^{2}(R^{m}) → L^{2}(R^{m}) such that (f, g) = (f̂, ĝ). (Plancherel theorem). We note that without the normalizing constant of (2π)^{−m/2} in the defi- nition of the measures dx and dξ there would be various normalizing con- stants appearing in these identities. It is property (b) which will be of the 6 1.1. Fourier Transform, Schwartz Class most interest to us since it will enable us to interchange differentiation and multiplication. We define the Sobolev space H_{s}(R^{m}) to measure L^{2} derivatives. If s is a real number and f ∈ S, we define: |f|_{2 s} = ∫ (1 + |ξ|^{2})^{s}|f̂(ξ)|^{2} dξ. The Sobolev space H_{s}(R^{m}) is the completion of S with respect to the norm |_{s}. The Plancherel theorem shows H_{0}(R^{m}) is isomorphic to L^{2}(R^{m}). More generally, H_{s}(R^{m}) is isomorphic to L^{2} with the measure (1 + |ξ|^{2})^{s/2} dξ. Replacing (1 + |ξ|^{2})^{s} by (1 + |ξ|)^{2s} in the definition of |_{s} gives rise to an equivalent norm since there exist positive constants c_{i} such that: c_{1}(1 + |ξ|^{2})^{s} ≤ (1 + |ξ|)^{2s} ≤ c_{2}(1 + |ξ|^{2})^{s}. In some sense, the subscript “s ” counts the number of L^{2} derivatives. If s = n is a positive integer, there exist positive constants c_{1}, c_{2} so: c_{1}(1 + |ξ|^{2})^{n} ≤ _{∑ |α|≤n} |ξ^{α}|^{2} ≤ c_{2}(1 + |ξ|^{2})^{n}. This implies that we could define |f|_{2 n} = _{∑ |α|≤n} ∫ |ξ^{α}f̂|^{2} dξ = _{∑ |α|≤n} ∫ |D_{α x}f|^{2} dx as an equivalent norm for H_{n}(R^{m}). With this interpretation in mind, it is not surprising that when we extend D_{α x} to H_{s}, that |α| L^{2} derivatives are lost. Lemma 1.1.3. D_{α x} extends to define a continuous map D_{α x}: H_{s} → H_{s−|α|}. Proof: Henceforth we will use C to denote a generic constant. C can depend upon certain auxiliary parameters which will usually be supressed in the interests of notational clarity. In this proof, for example, C depends on (s, α) but not of course upon f. The estimate: |ξ^{α}|^{2}(1 + |ξ|^{2})^{s−|α|} ≤ C(1 + |ξ|^{2})s implies that:|D_{α x}f|_{2 s−α} = ∫ |ξ^{α}f̂(ξ)|^{2}(1 + |ξ|^{2})^{s−|α|} dξ ≤ C|f|^{2} s for f ∈ S. Since H_{s} is the closure of S in the norm |_{s}, this completes the proof.

We can also use the sup norm to measure derivatives. If k is a non- negative integer, we define: |f|_{∞,k} = _{sup x∈Rm} _{∑ |α|≤k} |D_{α x}f|  for  f ∈ S . The completion of S with respect to this norm is a subset of C^{k}(R^{m}) (the continuous functions on R^{m} with continuous partial derivatives up to order k). The next lemma relates the two norms |_{s} and |_{∞,k}. It will play an important role in showing the weak solutions we will construct to differential equations are in fact smooth. Lemma 1.1.4. Let k be a non-negative integer and let s > k + _{m 2} . If f ∈ H_{s}, then f is C^{k} and there is an estimate |f|_{∞,k} ≤ C|f|_{s}. (Sobolev Lemma). Proof: Suppose first k = 0 and f ∈ S. We compute f(x) = ∫ e^{ix·ξ} f̂(ξ) dξ = ∫ {e^{ix·ξ} f̂(ξ)(1 + |ξ|^{2})^{s/2}} · {(1 + |ξ|^{2})^{−s/2}} dξ. We apply the Cauchy-Schwarz inequality to estimate: |f(x)|^{2} ≤ |f|_{2 s} ∫ (1 + |ξ|^{2})^{−s} dξ. Since 2s > m, (1+|ξ|^{2})^{−s} is integrable so |f(x)| ≤ C|f|_{s}. We take the sup over x ∈ R^{m} to conclude |f|_{∞,0} ≤ C|f|_{s} for f ∈ S. Elements of H_{s} are the limits in the |_{s} norm of elements of S. The uniform limit of continuous functions is continuous so the elements of H_{s} are continuous and the same norm estimate extends to H_{s}. If k > 0, we use the estimate: |D^{α x}f|^{∞,0} ≤ C|D^{α x}f|^{s−|α|} ≤ C|f|^{s}  for |α| ≤ k and s − k > m 2 to conclude |f|_{∞,k} ≤ C|f|_{s} for f ∈ S. A similar argument shows that the elements of H_{s} must be C^{k} and that this estimate continues to hold. If s > t, we can estimate (1 + |ξ|^{2})^{s} ≥ (1 + |ξ|^{2})^{t}. This implies that |f|_{s} ≥ |f|_{t} so the identity map on S extends to define an injection of H_{s} → H_{t} which is norm non-increasing. The next lemma shows that this injection is compact if we restrict the supports involved.

8 1.1. Fourier Transform, Schwartz Class Lemma 1.1.5. Let {f_{n}} ∈ S be a sequence of functions with support in a fixed compact set K. We suppose there is a constant C so |f_{n}|_{s} ≤ C for all n. Let s > t. There exists a subsequence f_{nk} which converges in H_{t}. (Rellich lemma). Proof: Choose g ∈ C_{0}(R^{m}) which is identically 1 on a neighborhood of K. Then gf^{n} = f^{n} so by Lemma 1.1.2(c) f̂n = ĝ ∗ f̂n. We let ∂^{j} = ∂ ∂ξj then ∂_{j}(ĝ ∗ f̂n) = ∂_{j}ĝ ∗ f̂n so that: |∂_{j}f̂n(ξ)| ≤ ∫ |{∂_{j}ĝ(ξ − ζ)} · f̂n(ζ)| dζ. We apply the Cauchy-Schwarz inequality to estimate: |∂_{j}f̂n(ξ)| ≤ |f_{n}|_{s} · {∫ |∂_{j}ĝ(ξ − ζ)|^{2}(1 + |ζ|^{2})^{−s} dζ}^{1/2} ≤ C · h(ξ) where h is some continuous function of ξ. A similar estimate holds for |f̂n(ξ)|. This implies that the {f̂n} form a uniformly bounded equi-con- tinuous family on compact ξ subsets. We apply the Arzela-Ascoli theorem to extract a subsequence we again label by f_{n} so that f̂n(ξ) converges uniformly on compact subsets. We complete the proof by verifying that fn converges in H_{t} for s > t. We compute: |f_{j} − f_{k}|_{2 t} = ∫ |f̂j − f̂k|^{2}(1 + |ξ|^{2})^{t} dξ. We decompose this integral into two parts, |ξ| ≥ r and |ξ| ≤ r. On |ξ| ≥ r we estimate (1 + |ξ|^{2})^{t} ≤ (1 + r^{2})^{t−s}(1 + |ξ|^{2})^{s} so that: ∫_{|ξ|≥r} |f̂j − f̂k|^{2}(1 + |ξ|^{2})^{t} dξ ≤ (1 + r^{2})^{t−s} ∫ |f̂j − f̂k|^{2}(1 + |ξ|^{2})^{s} dξ ≤ 2C(1 + r^{2})^{t−s}. If ε > 0 is given, we choose r so that 2C(1 + r^{2})^{t−s} < ε. The remaining part of the integral is over |ξ| ≤ r. The f̂j converge uniformly on compact subsets so this integral can be bounded above by ε if j, k > j(ε). This completes the proof. The hypothesis that the supports are uniformly bounded is essential. It is easy to construct a sequence {f_{n}} with |f_{n}|_{s} = 1 for all n and such that the supports are pair-wise disjoint. In this case we can find ε > 0 so that |f_{j} − f_{k}|_{t} > ε for all (j, k) so there is no convergent subsequence.

We fix ϕ ∈ S and let ϕ_{ε}(x) = ϕ(εx). We suppose ϕ(0) = 1 and fix f ∈ S. We compute: D_{α x}(f − ϕ_{ε}f) = (1 − ϕ_{ε})D_{α x}f + terms of the form ε^{j}D_{β x}ϕ(εx)D_{γ x}f. As ε → 0, these other terms go to zero in L^{2}. Since ϕ_{ε} → 1 pointwise, (1 − ϕ_{ε})D_{x}f goes to zero in L^{2}. This implies ϕ_{ε}f → f in H_{n} for any n ≥ 0 as ε → 0 and therefore ϕ_{ε}f → f in H_{s} for any s. If we take ϕ ∈ C_{∞ 0}(R^{m}), this implies C_{∞ 0}(R^{m}) is dense in H_{s} for any s. Each H_{s} space is a Hilbert space so it is isomorphic to its dual. Be- cause there is no preferred norm for H_{s}, it is useful to obtain an invariant alternative characterization of the dual space H_{∗ s}: Lemma 1.1.6. The L^{2} pairing which maps S × S → C extends to a map of H_{s} × H_{−s} → C which is a perfect pairing and which identifies H_{−s} with H_{∗ s}. That is: (a) |(f, g)| ≤ |f|_{s}|g|_{−s} for f, g ∈ S, (b) given f ∈ S there exists g ∈ S so (f, g) = |f|_{s}|g|_{−s} and we can define |f|_{s} = _{sup g∈S , g≠0} |(f, g)| |g|−s . Proof: This follows from the fact that H_{s} is L^{2} with the weight function (1+|ξ|^{2})^{s} and H_{−s} is L^{2} with the weight function (1+|ξ|^{2})^{−s}. We compute: (f, g) = (f̂, ĝ) = ∫ f̂(ξ)(1 + |ξ|^{2})^{s/2}ĝ(ξ)(1 + |ξ|^{2})^{−s/2} dξ and apply the Cauchy-Schwartz inequality to prove (a). To prove part (b), we note |f|_{s} ≥ _{sup g∈S , g≠0} |(f, g)| |g|−s . We take g to be defined by: ĝ = f̂(1 + |ξ|^{2})^{s} ∈ S and note that (f, g) = (f̂, ĝ) = |f|_{2 s} and that |g|_{2 −s} = |f|_{2 s} to see that equality can occur in (a) which proves (b) If s > t > u then we can estimate: (1 + |ξ|)^{2t} ≤ ε(1 + |ξ|)^{2s} + C(ε)(1 + |ξ|)2u for any ε > 0. This leads immediately to the useful estimate:

10 1.1. Fourier Transform, Schwartz Class Lemma 1.1.7. Let s > t > u and let ε > 0 be given. Then |f|_{t} ≤ ε|f|_{s} + C(ε)|f|_{u}. If V is a finite dimensional vector space, let C^{∞}(V) be the space of smooth complex valued maps of R^{m} → V. We choose a fixed Hermitian inner product on V and define S (V) and H_{s}(V) as in the scalar case. If dim(V) = k and if we choose a fixed orthonormal basis for V, then S (V) and H_{s}(V) become isomorphic to the direct sum of k copies of S and of H_{s}. Lemmas 1.1.1 through 1.1.7 extend in the obvious fashion. We conclude this subsection with an extremely useful if elementary esti- mate: Lemma 1.1.8. (Peetre’s Inequality). Let s be real and x, y ∈ R^{m}. Then (1 + |x + y|)^{s} ≤ (1 + |y|)^{s}(1 + |x|)^{|s|}. Proof: We suppose first s > 0. We raise the triangle inequality: 1 + |x + y| < 1 + |x| + |y| ≤ (1 + |y|)(1 + |x|) to the s^{th} power to deduce the desired inequality. We now suppose s < 0. A similar inequality: (1 + |y|)^{−s} ≤ (1 + |x + y|)^{−s}(1 + |x|)−s yields immediately:(1 + |x + y|)^{s} ≤ (1 + |y|)^{s}(1 + |x|)−s to complete the proof.

1.2. Pseudo-Differential Operators on R^{m}. A linear partial differential operator of order d is a polynomial expression P = p(x, D) = ∑_{|α|≤d} a_{α}(x)D_{α x} where the a_{α}(x) are smooth. The symbol σP = p is defined by: σP = p(x, ξ) = ^{∑ |α|≤d} a^{α}(x)ξα and is a polynomial of order d in the dual variable ξ. It is convenient to regard the pair (x, ξ) as defining a point of the cotangent space T^{∗}(R^{m}); we will return to this point again when we discuss the effect of coordinate transformations. The leading symbol σ_{L}P is the highest order part: σ^{L}P (x, ξ) = ^{∑ |α|=d} a^{α}(x)ξα and is a homogeneous polynomial of order d in ξ. We can use the Fourier inversion formula to express: P f(x) = ∫ e^{ix·ξ} p(x, ξ)f̂(ξ) dξ = ∫ e^{i(x−y)·ξ} p(x, ξ)f(y) dy dξ for f ∈ S. We note that since the second integral does not converge abso- lutely, we cannot interchange the dy and dξ orders of integration. We use this formalism to define the action of pseudo-differential operators (ΨDO’s) for a wider class of symbols p(x, ξ) than polynomials. We make the follow- ing Definition. p(x, ξ) is a symbol of order d and we write p ∈ S^{d} if (a) p(x, ξ) is smooth in (x, ξ) ∈ R^{m} × R^{m} with compact x support, (b) for all (α, β) there are constants C_{α,β} such that |D_{α x}D_{β ξ}p(x, ξ)| ≤ C_{α,β}(1 + |ξ|)^{d−|β|}. For such a symbol p, we define the associated operator P (x, D) by: P (x, D)(f)(x) = ∫ e^{ix·ξ} p(x, ξ)f̂(ξ) dξ = ∫ e^{(x−y)·ξ} p(x, ξ)f(y) dy dξ as a linear operator mapping S → S. A differential operator has as its order a positive integer. The order of a pseudo-differential operator is not necessarily an integer. For example, if f ∈ C_{∞ 0}(R^{m}), define: p(x, ξ) = f(x)(1 + |ξ|^{2})^{d/2} ∈ S^{d} for any d ∈ R.

12 1.2. Pseudo-Differential Operators This will be a symbol of order d. If p ∈ S^{d} for all d, then we say that p ∈ S^{−∞} is infinitely smoothing. We adopt the notational convention of letting p, q, r denote symbols and P, Q, R denote the corresponding ΨDO’s. Because we shall be interested in problems on compact manifolds, we have assumed the symbols have compact x support to avoid a number of technical complications. The reader should note that there is a well defined theory which does not require compact x support. When we discuss the heat equation, we shall have to consider a wider class of symbols which depend on a complex parameter. We postpone discussion of this class until later to avoid unnecessarily complicating the discussion at this stage. We shall phrase the theorems and proofs of this section in such a manner that they will generalize easily to the wider class of symbols. Our first task is to extend the action of P from S to H_{s}. Lemma 1.2.1. Let p ∈ S^{d} then |P f|_{s−d} ≤ C|f|_{s} for f ∈ S. P extends to a continuous map P: H_{s} → H_{s−d} for all s. Proof: We compute P f(x) = ∫ e^{ix·ξ} p(x, ξ)f̂(ξ) dξ so that the Fourier transform is given by: P f(ζ) = ∫ e^{ix·(ξ−ζ)}p(x, ξ)f̂(ξ) dξ dx. This integral is absolutely convergent since p has compact x support so we may interchange the order of integration. If we define q(ζ, ξ) = ∫ e^{−ix·ζ}p(x, ξ) dx as the Fourier transform in the x direction, then P f(ζ) = ∫ q(ζ − ξ, ξ)f̂(ξ) dξ. By Lemma 1.1.6, |P f|_{s−d} = _{sup g∈S} |(P f, g)| |g|d−s . We compute: (P f, g) = ∫ q(ζ − ξ, ξ)f̂(ξ)¯ĝ(ζ) dξ dζ. Define: K(ζ, ξ) = q(ζ − ξ, ξ)(1 + |ξ|)^{−s}(1 + |ζ|)s−d then: (P f, g) = ∫ K(ζ, ξ)f̂(ξ)(1 + |ξ|)^{s}¯ĝ(ζ)(1 + |ζ|)^{d−s} dξ dζ.

Operators on R^{m} 13

We apply the Cauchy-Schwarz inequality to estimate: |(P f, g)| ≤ {∫ |K(ζ, ξ)| |f̂(ξ)|^{2}(1 + |ξ|)^{2s} dξ dζ}1/2 × {∫ |K(ζ, ξ)| |ĝ(ζ)|^{2}(1 + |ζ|)^{2d−2s} dξ dζ}^{1/2} . We complete the proof by showing ∫ |K(ζ, ξ)| dξ ≤ C  and  ∫ |K(ζ, ξ)| dζ ≤ C since then |(P f, g)| ≤ C|f|_{s}|g|_{d−s}. By hypothesis, p has suppport in a compact set K and we have estimates: |D_{α x}p(x, ξ)| ≤ C_{α}(1 + |ξ|)^{d}. Therefore: |ζ^{α}q(ζ, ξ)| = ||||∫ e^{−ix·ζ}D_{α x}p(x, ξ) dx|||| ≤ C_{α}(1 + |ξ|)^{d} vol(K). Therefore, for any integer k, |q(ζ, ξ)| ≤ C_{k}(1 + |ξ|)^{d}(1 + |ζ|)^{−k} vol(K) and: |K(ζ, ξ)| ≤ C_{k}(1 + |ξ|)^{d−s}(1 + |ζ|)^{s−d}(1 + |ζ − ξ|)^{−k} vol(K). We apply Lemma 1.1.8 with x + y = ξ and y = ζ to estimate: |K(ζ, ξ)| ≤ C_{k}(1 + |ζ − ξ|)^{|d−s|−k} vol(K). If we choose k > _{m 2} + |d − s|, then this will be integrable and complete the proof. Our next task is to show that the class of ΨDO’s forms an algebra under the operations of composition and taking adjoint. Before doing that, we study the situation with respect to differential operators to motivate the formulas we shall derive. Let P = ∑_{α} p_{α}(x)D_{α x} and let Q = ∑_{α} q_{α}(x)D^{α} x be two differential operators. We assume p and q have compact x support. It is immediate that: P^{∗} = _{∑ α} D_{α x}p_{∗ α}  and P Q = _{∑ α,β} p_{α}(x)D_{α x}q_{β}(x)D^{β} x are again differential operators in our class. Furthermore, using Leibnitz’s rule  D_{α x}(fg) = _{∑ β+γ=α} D_{β x}(f) · D_{γ x}(g) · α! β!γ! , d_{β ξ}(ξ^{β+γ}) = ξ^{γ} · (β + γ)! γ! ,

it is an easy combinatorial exercise to compute that: σ(P^{∗}) = _{∑ α} d_{α ξ}D_{α x}p^{∗}/α!  and σ(P Q) = _{∑ α} d_{α ξ}p · D_{α x}q/α! . The perhaps surprising fact is that these formulas remain true in some sense for ΨDO’s, only the sums will become infinite rather than finite. We introduce an equivalence relation on the class of symbols by defining p ∼ q if p − q ∈ S^{−∞}. We note that if p ∈ S^{−∞} then P: H_{s} → H_{t} for all s and t by Lemma 1.2.1. Consequently by Lemma 1.1.4, P: H_{s} → C_{∞ 0} for all s so that P is infinitely smoothing in this case. Thus we mod out by infinitely smoothing operators. Given symbols p_{j} ∈ S^{dj} where d_{j} → −∞, we write p ∼ _{∞ ∑ j=1} pj if for every d there is an integer k(d) such that k ≥ k(d) implies that p − ∑_{k j=1} p_{j} ∈ S^{d}. We emphasize that this sum does not in fact need to converge. The relation p ∼ ∑ p_{j} simply means that the difference between P and the partial sums of the P_{j} is as smoothing as we like. It will turn out that this is the appropriate sense in which we will generalize the formulas for σ(P^{∗}) and σ(P Q) from differential to pseudo-differential operators. Ultimately, we will be interested in operators which are defined on com- pact manifolds. Consequently, it poses no difficulties to restrict the domain and the range of our operators. Let U be a open subset of R^{m} with com- pact closure. Let p(x, ξ) ∈ S^{d} have x support in U. We restrict the domain of the operator P to C_{∞ 0}(U) so P: C_{∞ 0}(U) → C_{∞ 0}(U). Let Ψ_{d}(U) denote the space of all such operators. For d ≤ d′, then Ψ_{d}(U) ⊆ Ψ_{d′}(U). We define  Ψ(U) = _{⋃ d} Ψ_{d}(U)  and Ψ_{−∞}(U) = _{⋂ d} Ψ_{d}(U) to be the set of all pseudo-differential operators on U and the set of infinitely smoothing pseudo-differential operators on U. More generally, let p(x, ξ) be a matrix valued symbol; we suppose the components of p all belong to S^{d}. The corresponding operator P is given by a matrix of pseudo-differential operators. P is a map from vector val- ued functions with compact support in U to vector valued functions with compact support in U. We shall not introduce separate notation for the shape of p and shall continue to denote the collection of all such operators by Ψ_{d}(U). If p and q are matrix valued and of the proper shape, we define p·q and also the operator P ·Q by matrix product and by composition. We also define p^{∗} and P^{∗} to be the matrix adjoint and the operator adjoint so that (P^{∗}f, g) = (f, P^{∗}g) where f and g are vector valued and of compact support. Before studying the algebra structure on Ψ(U), we must enlarge the class of symbols which we can admit:

Lemma 1.2.2. Let r(x, ξ, y) be a matrix valued symbol which is smooth in (x, ξ, y). We suppose r has compact x support inside U and that there are estimates:

|D^α_x D^β_ξ D^γ_y r| ≤ C_{α,β,γ} (1 + |ξ|)^{d−|β|}

for all multi-indices (α, β, γ). If f is vector valued with compact support in U, we define:

Rf(x) = ∫ e^{i(x−y)·ξ} r(x, ξ, y)f(y) dy dξ.

Then this operator is in Ψ_d(U) and the symbol is given by:

σR(x, ξ) ∼ {∑

α

d^α_ξ D^α_y r(x, ξ, y)/α!}|||||

x=y

.

Proof: We note that any symbol in S^d belongs to this class of operators if we define r(x, ξ, y) = p(x, ξ). We restricted to vector valued functions with compact support in U. By multiplying r by a cut-off function in y with compact support which is 1 over U, we may assume without loss of generality the y support of r is compact as well. Define:

q(x, ξ, ζ) = ∫ e^{−iy·ζ}r(x, ξ, y) dy

to be the Fourier transform of r in the y variable. Using Lemma 1.1.2 we see (rf) = r̂ ∗ f̂. This implies that: ∫

e^{−iy·ξ} r(x, ξ, y)f(y) dy = ∫ q(x, ξ, ξ − ζ)f̂(ζ) dζ.

The argument given in the proof of Lemma 1.2.1 gives estimates of the form:

|q(x, ξ, ζ)| ≤ C_k(1 + |ξ|)^d(1 + |ζ|)^{−k} and |f̂(ζ)| ≤ C_k(1 + |ζ|)^{−k}

for any k. Consequently:

|q(x, ξ, ξ − ζ)f̂(ζ)| ≤ C_k(1 + |ξ|)^d(1 + |ξ − ζ|)^{−k}(1 + |ζ|)^{−k}.

We apply Lemma 1.1.8 to estimate:

|q(x, ξ, ξ − ζ)f̂(ζ)| ≤ C_k(1 + |ξ|)^{|d|−k}(1 + |ζ|)^{|d|−k}

so this is absolutely integrable. We change the order of integration and express:  Rf(x) = ∫ e^{ix·ξ} q(x, ξ, ξ − ζ)f̂(ζ) dξ dζ. We define:  p(x, ζ) = ∫ e^{ix(ξ−ζ)}q(x, ξ, ξ − ζ) dξ and compute:  Rf(x) = ∫ e^{ix·ζ}p(x, ζ)f̂(ζ) dζ is a pseudo-differential operator once it is verified that p(x, ζ) is a symbol in the correct form. We change variables to express: p(x, ζ) = ∫ e^{ix·ξ} q(x, ξ + ζ, ξ) dξ and estimate: |q(x, ξ + ζ, ξ)| ≤ C^{k}(1 + |ξ + ζ|)^{d}(1 + |ξ|)−k ≤ C_{k}(1 + |ζ|)^{d}(1 + |ξ|)^{|d|−k}. This is integrable so |p(x, ζ)| ≤ C_{′k}(1 + |ζ|)^{d}. Similar estimates on |D_{α x}D^{β} ζ q(x, ξ + ζ, ξ)| which arise from the given estimates for r show that p ∈ S d so that R is a pseudo-differential operator. We use Taylor’s theorem on the middle variable of q(x, ξ+ζ, ξ) to expand: q(x, ξ + ζ, ξ) = _{∑ |α|≤k} d^{α ζ}q(x, ζ, ξ)ξα α!  + q_{k}(x, ζ, ξ). The remainder q_{k} decays to arbitrarily high order in (ξ, ζ) and after inte- gration gives rise to a symbol in S^{d−k} which may therefore be ignored. We integrate to conclude p(x, ζ) = _{∑ |α|≤k} ∫ e^{ix·ξ} d^{α ζ}q(x, ζ, ξ)ξα α!  dξ + remainder = _{∑ |α|≤k} d_{α ζ}D_{α y}r(x, ζ, y) α!  ||||_{x=y} + a remainder using Lemma 1.1.2. This completes the proof of the lemma. We use this technical lemma to show that the pseudo-differential opera- tors form an algebra:

Lemma 1.2.3. Let P ∈ Ψ_{d}(U) and let Q ∈ Ψ_{e}(U). Then: (a) If U′ is any open set with compact closure containing U, then P^{∗} ∈ Ψ_{d}(U′) and σ(P^{∗}) ∼ ∑_{α} d_{α ξ}D_{α x}p^{∗}/α! . (b) Assume that P and Q have the proper shapes so P Q and pq are defined. Then P Q ∈ Ψ_{d+e}(U) and σ(P Q) ∼ ∑_{α} d_{α ξ}p · D_{α x}q/α! . Proof: The fact that P^{∗} lies in a larger space is only a slight bit of technical bother; this fact plays an important role in considering boundary value problems of course. Let (f, g) = f · g be the pointwise Hermitian inner product. Fix ϕ ∈ C_{∞ 0}(U′) to be identically 1 on U and compute: (P f, g) = ∫ e^{i(x−y)·ξ} p(x, ξ)ϕ(y)f(y) · g(x) dy dξ dx = ∫ f(y) · e^{i(y−x)·ξ} p^{∗}(x, ξ)ϕ(y)g(x) dy dξ dx since the inner product is Hermitian. By approximating p^{∗}(x, ξ) by func- tions with compact ξ support, we can justify the use of Fubini’s theorem to replace dy dξ dx by dx dξ dy and express: (P f, g) = ∫ f(y) · e^{i(y−x)·ξ} p^{∗}(x, ξ)ϕ(y)g(x) dx dξ dy = (f, P^{∗}g) where we define:P^{∗}g(y) = ∫ e^{i(y−x)·ξ} p^{∗}(x, ξ)ϕ(y)g(x) dx dξ. This is an operator of the form discussed in Lemma 1.2.2 so P^{∗} ∈ Ψ_{d}(U′) and we compute: σ(P^{∗}) ∼ _{∑ α} d_{α ξ}D_{α x}p^{∗}/α! since ϕ = 1 on the support of p. This completes the proof of (a). We note that we can delete the factor of ϕ from the expression for P^{∗}g since it was only needed to prove P^{∗} was a ΨDO. We use (a) to prove (b). Since: Q^{∗}g(y) = ∫ e^{i(y−x)·ξ} q^{∗}(x, ξ)g(x) dx dξ the Fourier inversion formula implies: (Q^{∗}g) = ∫ e^{−ix·ξ} q^{∗}(x, ξ)g(x) dx.

18 1.2. Pseudo-Differential Operators If q̃ is the symbol of Q^{∗}, then we interchange the roles of Q and Q^{∗} to see: (Qg) = ∫ e^{−ix·ξ} q̃∗(x, ξ)g(x) dx. Therefore: P Qg(x) = ∫ e^{ix·ξ} p(x, ξ)(Qg)(ξ) dξ = ∫ e^{i(x−y)·ξ} p(x, ξ)q̃∗(y, ξ)g(y) dy dξ which is an operator of the form discussed in Lemma 1.2.2 if r(x, ξ, y) = p(x, ξ)q̃∗(y, ξ). This proves P Q is a pseudo-differential operator of the correct order. We compute the symbol of P Q to be: ∼ _{∑ α} d_{α ξ}D_{α y}(p(x, ξ)q̃∗(y, ξ))/α! evaluated at x = y. We use Leibnitz’s formula and expand this in the form: ∼ _{∑ β,γ} d_{β ξ}p(x, ξ)D_{β y}d_{γ ξ}D_{γ y}q̃∗/β!γ! . The sum over γ yields the symbol of Q^{∗∗} = Q so we conclude finally σ(P Q) ∼ _{∑ β} d_{β ξ}p(x, ξ)D_{β x}q(x, ξ)/β! which completes the proof. Let K(x, y) be a smooth matrix valued function with compact x support in U. If f is vector valued with compact support in U, we define: P (K)(f)(x) = ∫ K(x, y)f(y) dy. Lemma 1.2.4. Let K(x, y) be smooth with compact x support in U, then P (K) ∈ Ψ_{−∞}(U). Proof: We let ϕ(ξ) ∈ C_{∞ 0}(R^{m}) with ∫ ϕ(ξ) dξ = 1. Define: r(x, ξ, y) = e^{i(y−x)·ξ} ϕ(ξ)K(x, y) then this is a symbol in S^{−∞} of the sort discussed in Lemma 1.2.2. It defines an infinitely smoothing operator. It is immediate that: P (K)(f)(x) = ∫ e^{i(x−y)·ξ} r(x, ξ, y)f(y) dy dξ. Conversely, it can be shown that any infinitely smoothing map has a smooth kernel. In general, of course, it is not possible to represent an ar- bitrary pseudo-differential operator by a kernel. If P is smoothing enough, however, we can prove:

Operators on R^{m} 19

Lemma 1.2.5. Let r satisfy the hypothesis of Lemma 1.2.2 where d < −m − k. We define K(x, y) = ∫ e^{i(x−y)·ξ} r(x, ξ, y) dξ. Then K is C^{k} in (x, y) and Rf(x) = ∫ K(x, y)f(y) dy. Proof: If we can show K is well defined, then the representation of R in terms of the kernel K will follow from Fubini’s theorem. We estimate: D^{α x}D^{β y}K(x, y) = ^{∑ α=α_{1}+α2 β=β_{1}+β2} α! β! α_{1}! α_{2}! β_{1}! β_{2}!(−1)|β^{1}| × {∫ e^{i(x−y)·ξ} ξ^{α_{1}+β1} D_{α2 x} D_{β2 y} r(x, ξ, y) dξ} . Since we can estimate: |ξ^{α_{1}+β1} D^{α2 x} D^{β2 y} r(x, ξ, y)| ≤ C(1 + |ξ|)d+|α|+|β| this will be integrable for |α|+|β| ≤ k. Thus K is C^{k} and the representation of R follows immediately. In Lemma 1.2.2 we computed the symbol of the pseudo-differential op- erator defined by r(x, ξ, y) in terms of d_{α ξ}D_{α y}r when x = y. This implies the singular (i.e., the non-smoothing part) of R is concentrated near the diagonal x = y. We make this more precise: Lemma 1.2.6. Let r(x, ξ, y) satisfy the hypothesis of Lemma 1.2.2. Sup- pose the x support of r is disjoint from the y support of r, then R is infinitely smoothing and is represented by a smooth kernel function K(x, y). Proof: We would like to define K(x, y) = ∫ e^{(x−y)·ξ} r(x, ξ, y) dξ. Un- fortunately, this integral need not converge in general. By hypothesis, |x−y| ≥ ε > 0 on the support of r. We define the Laplacian Δ_{ξ} = ∑_{ν} D_{2 ξν}. Since Δ_{ξ}e^{i(x−y)·ξ} = |x−y|^{2}e^{i(x−y)·ξ} we integrate by parts in a formal sense k times to express: Rf(x) = ∫ e^{i(x−y)·ξ} |x − y|^{−2k}Δ_{k ξ}r(x, ξ, y)f(y) dy dξ. This formal process may be justified by first approximating r by a function with compact ξ support. We now define K(x, y) = ∫ e^{i(x−y)·ξ} |x − y|^{−2k}Δ_{k ξ}r(x, ξ, y) dξ for any k sufficiently large. Since Δ_{k ξ}r decays to arbitrarily high order in ξ, we use the same argument as that given in Lemma 1.2.5 to show that

K(x, y) is arbitrarily smooth in (x, y) and hence is C^{∞}. This completes the proof. We note that in general K(x, y) will become singular at x = y owing to the presence of the terms |x − y|^{−2k} if we do not assume the support of x is disjoint from the support of y. A differential operator P is local in the sense that if f = 0 on some open subset of U, then P f = 0 on that same subset since differentiation is a purely local process. ΨDO’s are not local in general since they are defined by the Fourier transform which smears out the support. Nevertheless, they do have a somewhat weaker property, they do not smear out the singular support of a distribution f. More precisely, let f ∈ H_{s}. If ϕ ∈ C_{∞ 0}(U), we define the map f ↦ ϕf. If we take r(x, ξ, y) = ϕ(x) and apply Lemma 1.2.2, then we see that this is a pseudo-differential operator of order 0. Therefore ϕf ∈ H_{s} as well. This gives a suitable notion of restriction. We say that f is smooth on an open subset U′ of U if and only if ϕf ∈ C^{∞} for every such ϕ. An operator P is said to be pseudo-local if f is smooth on U′ implies P f is smooth on U′. Lemma 1.2.7. Pseudo-differential operators are pseudo-local. Proof: Let P ∈ Ψ_{d}(U) and let f ∈ H_{s}. Fix x ∈ U′ and choose ϕ ∈ C_{∞ 0}(U′) to be identically 1 near x. Choose ψ ∈ C_{∞ 0}(U′) with support contained in the set where ϕ is identically 1. We must verify that ψP f is smooth. We compute: ψP f = ψP ϕf + ψP (1 − ϕ)f. By hypothesis, ϕf is smooth so ψP ϕf is smooth. The operator ψP (1 − ϕ) is represented by a kernel of the form ψ(x)p(x, ξ)(1 − ϕ(y)) which has disjoint x and y support. Lemma 1.2.6 implies ψP (1 − ϕ)f is smooth which completes the proof. In Lemmas 1.2.2 and 1.2.3 we expressed the symbol of an operator as a infinite asymptotic series. We show that the algebra of symbols is complete in a certain sense: Lemma 1.2.8. Let p_{j} ∈ S^{dj} (U) where d_{j} → −∞. Then there exists p ∼ ∑_{j} p_{j} which is a symbol in our class. p is a unique modulo S^{−∞}. Proof: We may assume without loss of generality that d_{1} > d_{2} > ⋯ → −∞. We will construct p ∈ S^{d1} . The uniqueness is clear so we must prove existence. The p_{j} all have support inside U; we will construct p with support inside U′ where U′ is any open set containing the closure of U. Fix a smooth function ϕ such that: 0 ≤ ϕ ≤ 1,  ϕ(ξ) = 0 for |ξ| ≤ 1,  ϕ(ξ) = 1 for |ξ| ≥ 2.

We use ϕ to cut away the support near ξ = 0. Let t_{j} → 0 and define: p(x, ξ) = _{∑ j} ϕ(t_{j}ξ)p_{j}(x, ξ). For any fixed ξ, ϕ(t_{j}ξ) = 0 for all but a finite number of j so this sum is well defined and smooth in (x, ξ). For j > 1 we have |p_{j}(x, ξ)| ≤ C_{j}(1 + |ξ|)^{dj} = C_{j}(1 + |ξ|)^{d1} (1 + |ξ|)^{d_{j}−d1} . If |ξ| is large enough, (1 + |ξ|)^{d_{j}−d1} is as small as we like and therefore by passing to a subsequence of the t_{j} we can assume |ϕ(t_{j}ξ)p_{j}(x, ξ)| ≤ 2^{−j}(1 − |ξ|)^{d1}  for j > 1. This implies that |p(x, ξ)| ≤ (C_{1} + 1)(1 + |ξ|)^{d1} . We use a similar ar- gument with the derivatives and use a diagonalization argument on the resulting subsequences to conclude p ∈ S^{d}. The supports of all the p_{j} are contained compactly in U so the support of p is contained in U which is contained in U′. We now apply exactly the same argument to p_{d2} + ⋯ to assume that p_{d2} + ⋯ ∈ S^{d2} . We continue in this fashion and use a diagonalization argument on the resulting subsequences to conclude in the end that_{∞ ∑ j=j0} ϕ(t_{j}ξ)p_{j}(x, ξ) ∈ S^{k}  for k = d_{j0}. Since p_{j} − ϕ(t_{j}ξ)p_{j} ∈ S^{−∞}, this implies p − ∑_{j0 j=1} p_{j} ∈ S^{k} and completes the proof. If K(x, y) is smooth with compact x, y support in U, then P (K) ∈ Ψ_{−∞}(U) defines a continuous operator from H_{s} → H_{t} for any s, t. Let |P |_{s,t} denote the operator norm so |P f|_{t} ≤ |P |_{s,t}|f|_{s} for any f ∈ S. It will be convenient to be able to estimate |K|_{∞,k} in terms of these norms: Lemma 1.2.9. Let K(x, y) be a smooth kernel with compact x, y support in U. Let P = P (K) be the operator defined by K. If k is a non-negative integer, then |K|_{∞,k} ≤ C(k)|P |−k,k Proof: By arguing separately on each entry in the matrix K, we may reduce ourselves to the scalar case. Suppose first k = 0. Choose ϕ ∈ C_{∞ 0}(R^{m}) positive with ∫ ϕ(x) dx = 1. Fix points (x_{0}, y_{0}) ∈ U × U and define:f_{n}(x) = n^{m}ϕ(n(x − x_{0}))  and g_{n}(y) = n^{m}ϕ(n(y − y_{0})).

22 1.2. Pseudo-Differential Operators Then if n is large, f_{n} and g_{n} have compact support in U. Then: K(x_{0}, y_{0}) = _{lim n→∞} ∫ f_{n}(x)K(x, y)g_{n}(y) dy dx = _{lim n→∞}(f_{n}, P g_{n}) by Lemma 1.1.1. We estimate |(f_{n}, P g_{n})| ≤ |P |_{0,0}|f_{n}|_{0}|g_{n}|_{0} = |P |_{0,0}|ϕ|^{2} 0 to complete the proof in this case. If |α| ≤ k, |β| ≤ k then: D_{α x}D_{β y}K(x, y) = _{lim n→∞} ∫ f_{n}(x){D_{α x}D_{β y}K(x, y)}g_{n}(y) dy dx = _{lim n→∞} ∫ (D_{α x}f_{n})K(x, y)(D_{β y}g_{n}(y)) dy dx = _{lim n→∞}(D_{α x}f_{n}, P D_{β y}g_{n}). We use Lemma 1.1.6 to estimate this by |D_{α x}f_{n}|_{−k}|P D_{β y}g_{n}|k ≤ |f_{n}|_{0}|P |_{−k,k}|D_{β y}g_{n}|−k ≤ |f_{n}|_{0}|g_{n}|_{0}|P |_{−k,k} = |P |_{−k,k}|ϕ|^{2} 0 to complete the proof.

1.3. Ellipticity and Pseudo-Differential Operators on Manifolds. The norms we have given to define the spaces H_{s} depend upon the Fourier transform. In order to get a more invariant definition which can be used to extend these notions to manifolds, we must consider elliptic pseudo- differential operators. Let p ∈ S^{d}(U) be a square matrix and let U_{1} be an open set with U_{1} ⊂ U. We say that p is elliptic on U_{1} if there exists an open subset U2 with U^{1} ⊂ U^{2} ⊂ U^{2} ⊂ U and if there exists q ∈ S^{−d} such that pq−I ∈ S −∞ and qp−I ∈ S^{−∞} over U_{2}. (To say that r ∈ S^{−∞} over U_{2} simply means the estimates of section 1.2 hold over U^{2}. Equivalently, we assume ϕr ∈ S −∞ for every ϕ ∈ C_{∞ 0}(U_{2})). This constant technical fuss over domains will be eliminated very shortly when we pass to considering compact manifolds; the role of U_{2} is to ensure uniform estimates over U_{1}. It is clear that p is elliptic over U_{1} if and only if there exists constants C_{0} and C_{1} such that p(x, ξ) is invertible for |ξ| ≥ C_{0} and |p(x, ξ)^{−1}| ≤ C_{1}(1 + |ξ|)^{−d} for |ξ| ≥ C_{0}, x ∈ U_{2}. We define q = ϕ(ξ)p^{−1}(x, ξ) where ϕ(ξ) is a cut-off function identically 0 near ξ = 0 and identically 1 near ξ = ∞. We used similar cutoff functions in the proof of Lemma 1.2.8. Furthermore, if p_{0} ∈ S^{d−1}, then p is elliptic if and only if p + p_{0} is elliptic; adding lower order terms does not alter the ellipticity. If p is a polynomial and P is a differential operator, then p is elliptic if and only if the leading symbol σ_{L}(p) = ∑_{|α|=d} p_{α}(x)ξ^{α} is invertible for ξ ≠ 0. There exist elliptic operators of all orders. Let ϕ(x) ∈ C_{∞ 0} and define the symbol p(x, ξ) = ϕ(x)(1 + |ξ|^{2})^{d/2}I, then this is an elliptic symbol of order d whenever ϕ(x) ≠ 0. Lemma 1.3.1. Let P ∈ Ψ^{d}(U) be elliptic over U_{1} then: (a) There exists Q ∈ Ψ^{−d}(U) such that P Q − I ∼ 0 and QP − I ∼ 0 over U_{1} (i.e., ϕ(P Q − I) and ϕ(QP − I) are infinitely smoothing for any ϕ ∈ C_{∞ 0}(U_{2})). (b) P is hypo-elliptic over U_{1}, i.e., if f ∈ H_{s} and if P f is smooth over U1 then f is smooth over U_{1}. (c) There exists a constant C such that |f|_{d} ≤ C(|f|_{0} + |P f|_{0}) for f ∈ C_{∞ 0}(U_{1}). (Gärding’s inequality). Proof: We will define Q to have symbol q_{0} + q_{1} + ⋯ where q_{j} ∈ S^{−d−j}. We try to solve the equation σ(P Q − I) ∼ _{∑ α,j} d_{α ξ}p · D_{α x}q_{j}/α! − I ∼ 0.

When we decompose this sum into elements of S^{−k}, we conclude we must solve  ^{∑ |α|+j=k} d^{α ξ}p · D^{α x}q^{j}/α! = { I  if k = 0 0  if k ≠ 0. We define q_{0} = q and then solve the equation inductively to define: q_{k} = −q · _{∑ |α|+j=k j<k} d_{α ξ}p · D_{α x}q_{j}/α! . This defines Q so σ(P Q−I) ∼ 0 over U_{2}. Similarly we could solve σ(Q_{1}P − I) ∼ 0 over U_{2}. We now compute σ(Q − Q_{1}) = σ(Q − Q_{1}P Q) + σ(Q_{1}P Q − Q_{1}) = σ((I − Q_{1}P)Q) + σ(Q_{1}(QP − I)) ∼ 0 over U_{2} so that in fact Q and Q_{1} agree modulo infinitely smoothing operators. This proves (a). Let f ∈ H_{s} with P f smooth over U_{1}, and choose ϕ ∈ C_{0}(U_{1}). We compute:  ϕf = ϕ(I − QP)f + ϕQP f. As ϕ(I − QP) ∼ 0, ϕ(I − QP)f is smooth. Since P f is smooth over U_{1}, ϕQP f is smooth since Q is pseudo-local. Thus ϕf is smooth which proves (b). Finally, we choose ϕ ∈ C_{∞ 0}(U_{2}) to be identically 1 on U_{1}. Then if f ∈ C_{∞ 0}(U_{1}), |f|_{d} = |ϕf|_{d} = |ϕ(I − QP)f + ϕQP f|_{d} ≤ |ϕ(I − QP)f|_{d} + |ϕQP f|_{d}. We estimate the first norm by C|f|_{0} since ϕ(I − QP) is an infinitely smoothing operator. We estimate the second norm by C|P f|_{0} since ϕQ is a bounded map from L^{2} to H^{d}. This completes the proof. We note (c) is immediate if d < 0 since |_{d} ≤ |_{0}. If d > 0, |f|_{0} + |P f|_{0} ≤ C(|f|_{d}) so this gives a equivalent norm on H^{d}. We now consider the effect of changes of coordinates on our class of pseudo-differential operators. Let h: U → U be a diffeomorphism. We define h^{∗}: C^{∞}(U) → C^{∞}(U) by h^{∗}f(x) = f(h(x)). If P is a linear operator on C^{∞}(U), we define h_{∗}P acting on C^{∞}(U) by (h_{∗}P)f = (h^{−1})^{∗}P (h^{∗}f). The fundamental lemma we shall need is the following: Lemma 1.3.2. Let h: U → U be a diffeomorphism. Then: (a) If P ∈ Ψ^{d}(U) then h_{∗}P ∈ Ψ^{d}(U). Let p = σ(P) and define h(x) = x1 and dh(x)^{t}ξ_{1} = ξ. Let p_{1}(x_{1}, ξ_{1}) = p(x, ξ) then σ(h_{∗}P) − p_{1} ∈ S^{d−1}. (b) Let U_{1} be an open subset with U_{1} ⊂ U. There exists a constant C such that |h^{∗}f|_{d} ≤ C|f|_{d} for all f ∈ C_{∞ 0}(h(U_{1})). In other words, the Sobolev spaces are invariant. Proof: The first step is to localize the problem. Let {ϕ_{i}} be a partition of unity and let P_{ij} = ϕ_{i}P ϕ_{j} so P = ∑_{i,j} P_{ij}. If the support of ϕ_{i} is

disjoint from the support of ϕ_{j}, then P_{ij} is an infinitely smoothing operator with a smooth kernel K_{ij}(x, y) by Lemma 1.2.6. Therefore h_{∗}P_{ij} is also given by a smooth kernel and is a pseudo-differential operator by Lemma 1.2.4. Consequently, we may restrict attention to pairs (i, j) such that the supports of ϕ_{i} and ϕ_{j} intersect. We assume henceforth P is defined by a symbol p(x, ξ, y) where p has arbitrarily small support in (x, y). We first suppose h is linear to motivate the constructions of the general case. Let h(x) = hx where h is a constant matrix. We equate: hx = x_{1},  hy = y_{1},  h^{t}ξ_{1} = ξ and define  p_{1}(x_{1}, ξ_{1}, y_{1}) = p(x, ξ, y). (In the above, h^{t} denotes the matrix transpose of h). If f ∈ C_{∞ 0}(U), we compute:(h_{∗}P)f(x_{1}) = ∫ e^{i(x−y)·ξ} p(x, ξ, y)f(hy) dy dξ = ∫ e^{ih^{−1}(x_{1}−y_{1})·ξ} p(h^{−1}x_{1}, ξ, h^{−1}y_{1})f(y_{1}) × | det(h)|^{−1} dy_{1} dξ. We now use the identities h^{−1}(x_{1} −y_{1})·ξ = (x_{1} −y_{1})·ξ_{1} and | det(h)| dξ_{1} = dξ to write: (h_{∗}P)f(x_{1}) = ∫ e^{i(x_{1}−y_{1})·ξ1} p(h^{−1}x_{1}, h^{t}ξ_{1}, h^{−1}y_{1})f(y_{1}) dy_{1} dξ1 = ∫ e^{i(x_{1}−y_{1})·ξ1} p_{1}(x_{1}, ξ_{1}, y_{1})f(y_{1}) dy_{1} dξ_{1}. This proves that (h_{∗}P) is a pseudo-differential operator on U. Since we don’t need to localize in this case, we compute directly that σ(h_{∗}P)(x_{1}, ξ_{1}) = p(h^{−1}x_{1}, h^{t}ξ_{1}). We regard (x, ξ) as giving coordinates for T^{∗}M when we expand any co- vector in the form ∑ ξ_{i} dx^{i}. This is exactly the transformation for the cotangent space so we may regard σP as being invariantly defined on T^{∗}R^{m}. If h is not linear, the situation is somewhat more complicated. Let x − y = h^{−1}(x_{1}) − h^{−1}(y_{1}) = ∫_{1 0} d dt{h^{−1}(tx_{1} + (1 − t)y_{1})} dt = ∫_{1 0} d(h^{−1})(tx^{1} + (1 − t)y^{1}) · (x^{1} − y^{1}) dt = T (x^{1}, y^{1})(x^{1} − y^{1})

26 1.3. Ellipticity and Pseudo-Differential where T (x_{1}, y_{1}) is a square matrix. If x_{1} = y_{1}, then T (x_{1}, y_{1}) = d(h^{−1}) is invertible since h is a diffeomorphism. We localize using a partition of unity to suppose henceforth the supports are small enough so T (x_{1}, y_{1}) is invertible for all points of interest. We set ξ_{1} = T (x_{1}, y_{1})^{t}ξ and compute: (h_{∗}P)(f)(x_{1}) = ∫ e^{i(x−y)·ξ} p(x, ξ, y)f(hy) dy dξ = ∫ e^{iT (x_{1},y_{1})(x_{1}−y_{1})·ξ} p(h^{−1}x_{1}, ξ, h^{−1}y_{1})f(y_{1})J dy_{1} dξ = ∫ e^{i(x_{1}−y_{1})·ξ1} p_{1}(x_{1}, ξ_{1}, y_{1})f(y_{1}) × J| det T (x_{1}, y_{1})|^{−1} dy_{1} dξ1 where J = | det(dh^{−1})| = | det T (y_{1}, y_{1})|. By Lemma 1.2.2, this defines a pseudo-differential operator of order d such that σ(h^{∗}P) = p^{1} modulo S d−1 which completes the proof of (a). Since |dh| is uniformly bounded on U_{1}, |f|_{0} ≤ C|h^{∗}f|_{0} and |h^{∗}f|_{0} ≤ C|f|_{0}. If P is elliptic, then h_{∗}P is elliptic of the same order. For d > 0, choose P elliptic of order d and compute: |h^{∗}f|_{d} ≤ C(|h^{∗}f|_{0} + |P h^{∗}f|_{0}) ≤ C(|f|_{0} + |(h_{∗}P)f|_{0}) ≤ C|f|d which completes the proof of (b) if d ≥ 0. The result for d ≤ 0 follows by duality using Lemma 1.1.6(b). We introduce the spaces S^{d}/S^{d−1} and define σ_{L}(P) to be the element defined by σ(P) in this quotient. Let P and Q be pseudo-differential op- erators of order d_{1} and d_{2} Then P Q is a pseudo-differential operator of order d_{1} + d_{2} and σ_{L}(P Q) = σ_{L}(P)σ_{L}(Q) since the remaining terms in the asymptotic series are of lower order. Similarly σ_{L}(P^{∗}) = σ_{L}(P)^{∗}. If we define (x, ξ) as coordinates for T^{∗}(R^{m}) by representing a cotangent vector at a point x in the form ∑ ξ_{i}dx^{i}, then Lemma 1.3.2 implies σ_{L}(P) is invari- antly defined on T^{∗}(R^{m}). If P = ∑_{α} p_{α}D_{α x} is a differential operator there is a natural identification of ∑^{|α|=d} p^{α}ξ^{α} with the image of p in S^{d}/S d−1 so this definition of the leading symbol agrees with that given earlier. We now extend the results of section 1.2 to manifolds. Let M be a smooth compact Riemannian manifold without boundary. Let m be the dimension of M and let dvol or sometimes simply dx denote the Riemanian measure on M. In Chapter 2, we will use the notation | dvol | to denote this measure in order to distinguish between measures and m-forms, but we shall not bother with this degree of formalism here. We restrict to scalars first. Let C^{∞}(M) be the space of smooth functions on M and let P: C^{∞}(M) → C^{∞}(M) be a linear operator. We say that P is a pseudo- differential operator of order d and write P ∈ Ψ_{d}(M) if for every open chart

U on M and for every ϕ, ψ ∈ C_{∞ 0}(U), the localized operator ϕP ψ ∈ Ψ_{d}(U). We say that P is elliptic if ϕP ψ is elliptic where ϕψ(x) ≠ 0. If Q ∈ Ψ_{d}(U), we let P = ϕQψ for ϕ, ψ ∈ C_{∞ 0}(U). Lemma 1.3.2 implies P is a pseudo- differential operator on M so there exists operators of all orders on M. We define: Ψ(M) = _{⋃ d} Ψ_{d}(M)  and Ψ_{−∞}(M) = _{⋂ d} Ψ_{d}(M) to be the set of all pseudo-differential operators on M and the set of in- finitely smoothing operators on M. In any coordinate system, we define σ(P) to the symbol of the operator ϕP ϕ where ϕ = 1 near the point in question; this is unique modulo S^{−∞}. The leading symbol is invariantly defined on T^{∗}M, but the total symbol changes under the same complicated transformation that the total symbol of a differential operator does under coordinate transformations. Since we shall not need this transformaton law, we omit the statement; it is implicit in the computations performed in Lemma 1.3.2. We define L^{2}(M) using the L^{2} inner product (f, g) = ∫_{M} f(x)g(x) dx,  |f|_{2 0} = (f, f). We let L^{2}(M) be the completion of C^{∞}(M) in this norm. Let P: C^{∞}(M) → C^{∞}(M). We let P^{∗} be defined by (P f, g) = (f, P^{∗}g) if such a P^{∗} exists. Lemmas 1.3.2 and 1.2.3 imply that: Lemma 1.3.3. (a) If P ∈ Ψ_{d}(M), then P^{∗} ∈ Ψ_{d}(M) and σ_{L}(P^{∗}) = σ_{L}(P)^{∗}. In any coor- dinate chart, σ(P^{∗}) has a asymptotic expansion given by Lemma 1.2.3(a). (b) If P ∈ Ψ_{d}(M) and Q ∈ Ψ_{e}(M), then P Q ∈ Ψ_{d+e}(M) and σ_{L}(P Q) = σ_{L}(P)σ_{L}(Q). In any coordinate chart σ(P Q) has an asymptotic expansion given in Lemma 1.2.3(b). We use a partition of unity to define the Sobolev spaces H_{s}(M). Cover M by a finite number of coordinate charts U_{i} with diffeomorphisms h_{i}: O_{i} → U_{i} where the O_{i} are open subsets of R^{m} with compact closure. If f ∈ C_{∞ 0}(U_{i}), we define:  |f|_{(i) s} = |h_{∗ i}f|s where we shall use the superscript ^{(i)} to denote the localized norm. Let {ϕ_{i}} be a partition of unity subordinate to this cover and define: |f|^{s} = ∑ |ϕ^{i}f|_{(i) s}.

If ψ ∈ C^{∞}(M), we note that |ϕ_{i}ψf|_{(i) s} ≤ C|ϕ_{i}f|_{(i) s} since multiplication by ψ defines a ΨDO of order 0. Suppose {U_{′j}, O_{′j}, h_{′j}, ϕ_{′j}} is another possible choice to define |_{′s}. We estimate: |ϕ_{′j}f|_{′(j) s} ≤ _{∑ i} |ϕ_{′j}ϕ_{i}f|_{′(j) s}. Since ϕ_{′j}ϕ_{i}f ∈ C_{∞ 0}(U_{i} ∩ U_{′j}), we can use Lemma 1.3.2 (b) to estimate |^{′(j)} s by |_{(i) s} so  |ϕ_{′j}f|_{′(j) s} ≤ _{∑ i} C|ϕ_{′j}ϕ_{i}f|_{(i) s} ≤ C|ϕ_{i}f|_{(i) s} ≤ C|f|s so that |f|_{′s} ≤ C|f|_{s}. Similarly |f|_{s} ≤ C|f|_{′s}. This shows these two norms are equivalent so H_{s}(M) is defined independent of the choices made. We note that: ^{∑ i}{|ϕ^{i}f|^{(i) s}}^{2} ≤ |f|^{2 s} ≤ C^{∑ i}{|ϕ^{i}f|^{(i) s}}2 so by using this equivalent norm we conclude the H_{s}(M) are topologically Hilbert spaces. If {ψ_{i}} are given subordinate to the cover U_{i} with ψ_{i} ≥ 0 and ψ = ∑_{i} ψ_{i} > 0, we let ϕ_{i} = ψ_{i}/ψ and compute:_{∑ i} |ψ_{i}f|_{(i) s} = _{∑ i} |ψϕ_{i}f|_{(i) s} ≤ C_{∑ i} |ϕ_{i}f|_{(i) s} = C|f|s_{∑ i} |ϕ_{i}f|_{(i) s} = _{∑ i} |ψ^{−1}ψ_{i}f|_{(i) s} ≤ C_{∑ i} |ψ_{i}f|^{(i)} s to see the norm defined by ∑_{i} |ψ_{i}f|_{(i) s} is equivalent to the norm |f|_{s} as well. Lemma 1.1.5 implies that |f|_{t} ≤ |f|_{s} if t < s and that the inclusion of H_{s}(M) → H_{t}(M) is compact. Lemma 1.1.7 implies given s > t > u and ε > 0 we can estimate: |f|_{t} ≤ ε|f|_{s} + C(ε)|f|_{u}. We assume the coordinate charts U_{i} are chosen so the union U_{i} ∪ U_{j} is also contained in a larger coordinate chart for all (i, j). We decompose p ∈ Ψ_{d}(M) as P = ∑_{i,j} P_{i,j} for P_{i,j} = ϕ_{i}P ϕ_{j}. By Lemma 1.2.1 we can estimate:  |ϕ_{i}P ϕ_{j}f|_{(i) s} ≤ C|ϕ_{j}f|^{(j)} s+d so |P f|_{s} ≤ C|f|_{s+d} and P extends to a continuous map from H_{s+d}(M) → H_{s}(M) for all s.

We define:  |f|_{∞,k} = _{∑ i} |ϕ_{i}f|^{(i)} ∞,k as a measure of the sup norm of the k^{th} derivatives of f. This is independent of particular choices made. Lemma 1.1.4 generalizes to: |f|_{∞,k} ≤ C|f|_{s}  for s > m 2 + k. Thus H_{s}(M) is a subset of C^{k}(M) in this situation. We choose ψ_{i} ∈ C^{∞}(U_{i}) with ∑_{i} ψ_{2 i} = 1 then: |(f, g)| = |_{∑ i}(ψ_{i}f, ψ_{i}g)| ≤ _{∑ i} |(ψ_{i}f, ψ_{i}g)| ≤ C_{∑ i} |ψ_{i}f|_{(i) s}|ψ_{i}g|_{(i) −s} ≤ C|f|_{s}|g|_{−s}. Thus the L^{2} inner product gives a continuous map H_{s}(M)×H_{−s}(M) → C. Lemma 1.3.4. (a) The natural inclusion H_{s} → H_{t} is compact for s > t. Furthermore, if s > t > u and if ε > 0, then |f|_{t} ≤ ε|f|_{s} + C(ε)|f|_{u}. (b) If s > k + _{m 2} then H_{s}(M) is contained in C^{k}(M) and we can estimate |f|_{∞,k} ≤ C|f|_{s}. (c) If P ∈ Ψ_{d}(M) then P: H_{s+d}(M) → H_{s}(M) is continuous for all s. (d) The pairing H_{s}(M) × H_{−s}(M) → C given by the L^{2} inner product is a perfect pairing. Proof: We have proved every assertion except the fact (d) that the pair- ing is a perfect pairing. We postpone this proof briefly until after we have discussed elliptic ΨDO’s. The sum of two elliptic operators need not be elliptic. However, the sum of two elliptic operators with positive symbols is elliptic. Let P_{i} have symbol (1+|ξ_{i}|^{2})^{d/2} on U_{i} and let ϕ_{i} be a partition of unity. P = ∑_{i} ϕ_{i}P_{i}ϕ_{i}; this is an elliptic ΨDO of order d for any d, so elliptic operators exist. We let P be an elliptic ΨDO of order d and let ψ_{i} be identically 1 on the support of ϕ_{i}. We use these functions to construct Q ∈ Ψ_{−d}(M) so P Q − I ∈ Ψ_{−∞}(M) and QP − I ∈ Ψ_{−∞}(M). In each coordinate chart, let P_{i} = ψ_{i}P ψ_{i} then P_{i} − P_{j} ∈ Ψ_{−∞} on the support of ϕ_{i}ϕ_{j}. We construct Qi as the formal inverse to P_{i} on the support of ϕ_{i}, then Q_{i} − Q_{j} ∈ Ψ_{−∞} on the support of ϕ_{i}ϕ_{j} since the formal inverse is unique. Modulo Ψ_{−∞} we have P = ∑_{i} ϕ_{i}P ∼ ∑_{i} ϕ_{i}P_{i}. We define Q = ∑_{j} Q_{j}ϕ_{j} and note Q has the desired properties. It is worth noting we could also construct the formal inverse using a Neumann series. By hypothesis, there exists q so qp−I ∈ S^{−1} and pq −I ∈

30 1.3. Ellipticity and Pseudo-Differential Operators S^{−1}, where p = σ_{L}(P). We construct Q_{1} using a partition of unity so σ_{L}(Q_{1}) = q. We let Q^{r} and Q^{l} be defined by the formal series: Ql = Q^{1}{^{∑ k}(−1)^{k}(P Q^{1} − I)^{k}} Qr = {_{∑ k}(−1)^{k}(Q_{1}P − I)^{k}}Q1 to construct formal left and right inverses so Q = Q^{r} = Q^{l} modulo Ψ_{−∞}. Let P be elliptic of order d > 0. We estimate: |f|_{d} ≤ |(QP − I)f|_{d} + |QP f|_{d} ≤ C|f|_{0} + C|P f|_{0} ≤ C|f|d so we could define H_{d} using the norm |f|_{d} = |f|_{0} + |P f|_{0}. We specialize to the following case. Let Q be elliptic of order d/2 and let P = Q^{∗}Q + 1. Then Q^{∗}Q is self-adjoint and non-negative so we can estimate |f|_{0} ≤ |P f|0 and we can define |f|_{d} = |P f|_{0} in this case. Consequently: |f|_{2 d} = (P f, P f) = (f, P^{∗}P f) = (f, g), for g = P^{∗}P f. Since |P^{∗}P f|_{−d} ≤ C|f|_{d}, we conclude: |f|_{d} = (P f, P f)/|f|_{d} ≤ C(f, g)/|g|_{−d} ≤ C_{sup h} |(f, h)|/|h|_{−d}. Since the pairing of H_{d} with H_{−d} is continuous, this proves H_{d} = H_{∗ −d}. Topologically these are Hilbert spaces so we see dually that H_{∗ d} = H_{−d}. This completes the proof of Lemma 1.3.4. We also note that we have proved: Lemma 1.3.5. Let P ∈ Ψ_{d} be elliptic. Then there exists Q ∈ Ψ_{−d} so P Q − I ∈ Ψ_{−∞} and QP − I ∈ Ψ_{−∞}. P is hypoelliptic. If d > 0, we can define h_{d} by using the norm |f|_{0} + |P f|_{0} and define H_{−d} by duality. If V is a vector bundle, we cover M by coordinate charts U_{i} over which V is trivial. We use this cover to define H_{s}(V) using a partition of unity. We shall always assume V has a given fiber metric so L^{2}(V) is invariantly defined. P: C^{∞}(V) → C^{∞}(W) is a ΨDO of order d if ϕP ψ is given by a matrix of d^{th} order ΨDO’s for ϕ, ψ ∈ C_{∞ 0}(U) for any coordinate chart U over which V and W are trivial. Lemmas 1.3.4 and 1.3.5 generalize immediately to this situation.

1.4. Fredholm Operators and the Index of a Fredholm Operator. Elliptic ΨDO’s are invertible modulo Ψ_{−∞}. Lemma 1.3.4 will imply that elliptic ΨDO’s are invertible modulo compact operators and that such op- erators are Fredholm. We briefly review the facts we shall need concerning Fredholm and compact operators. Let H be a Hilbert space and let END(H) denote the space of all bounded linear maps T: H → H. There is a natural norm on END(H) defined by: |T | = ^{sup x∈H} |T x| |x| where the sup ranges over x ≠ 0. END(H) becomes a Banach space under this norm. The operations of addition, composition, and taking adjoint are continuous. We let GL(H) be the subset of END(H) consisting of maps T which are 1-1 and onto. The inverse boundedness theorem shows that if T ∈ END(H) is 1-1 and onto, then there exists ε > 0 such that |T x| ≥ ε|x| so T^{−1} is bounded as well. The Neuman series: (1 − z)^{−1} = ^{∞ ∑ k=0} zk converges for |z| < 1. If |I − T | < 1, we may express T = I − (I − T). If we define S = ^{∞ ∑ k=0}(I − T)k then this converges in END(H) to define an element S ∈ END(H) so ST = T S = I. Furthermore, this shows |T |^{−1} ≤ (1 − |I − T |)^{−1} so GL(H) contains an open neighborhood of I and the map T → T^{−1} is continuous there. Using the group operation on GL(H), we see that GL(H) is a open subset of END(H) and is a topological group. We say that T ∈ END(H) is compact if T maps bounded sets to pre- compact sets—i.e., if |x_{n}| ≤ C is a bounded sequence, then there exists a subsequence x_{nk} so T x_{nk} → y for some y ∈ H. We let COM(H) denote the set of all compact maps. Lemma 1.4.1. COM(H) is a closed 2-sided ∗-ideal of END(H). Proof: It is clear the sum of two compact operators is compact. Let T ∈ END(H) and let C ∈ COM(H). Let {x_{n}} be a bounded sequence in H then {T x_{n}} is also a bounded sequence. By passing to a subsequence, we may assume Cx_{n} → y and CT x_{n} → z. Since T Cx_{n} → T y, this implies CT and T C are compact so COM(H) is a ideal. Next let C_{n} → C in END(H), and let x_{n} be a bounded sequence in H. Choose a subsequence 32 1.4. Fredholm Operators x_{1 n} so C_{1}x_{1 n} → y^{1}. We choose a subsequence of the x_{1 n} so C_{2}x_{2 n} → y^{2}. By continuing in this way and then using the diagonal subsequence, we can find a subsequence we denote by x_{n n} so C_{k}(x_{n n}) → y^{k} for all k. We note |Cx_{n n} − C_{k}x_{n n}| ≤ |C − C_{k}|c. Since |C − C_{k}| → 0 this shows the sequence Cx_{n n} is Cauchy so C is compact and COM(H) is closed. Finally let C ∈ COM(H) and suppose C^{∗} / ∈ COM(H). We choose |x_{n}| ≤ 1 so |C^{∗}x_{n} − C^{∗}x_{m}| ≥ ε > 0 for all n, m. We let y_{n} = C^{∗}x_{n} be a bounded sequence, then (Cy_{n} − Cy_{m}, x_{n} − x_{m}) = |C^{∗}x_{n} − C^{∗}x_{m}|^{2} ≥ ε^{2}. Therefore ε^{2} ≤ |Cy_{n} − Cy_{m}||x_{n} − x_{m}| ≤ 2|Cy_{n} − Cy_{m}| so Cy_{n} has no convergent subsequence. This contradicts the assumption C ∈ COM(H) and proves C^{∗} ∈ COM(H). We shall assume henceforth that H is a separable infinite dimensional space. Although any two such Hilbert spaces are isomorphic, it is conven- ient to separate the domain and range. If E and F are Hilbert spaces, we define HOM(E, F) to be the Banach space of bounded linear maps from E to F with the operator norm. We let ISO(E, F) be the set of invert- ible maps in HOM(E, F) and let COM(E, F) be the closed subspace of HOM(E, F) of compact maps. If we choose a fixed isomorphism of E with F, we may identify HOM(E, F) = END(E), ISO(E, F) = GL(E), and COM(E, F) = COM(E). ISO(E, F) is a open subset of HOM(E, F) and the operation of taking the inverse is a continuous map from ISO(E, F) to ISO(F, E). If T ∈ HOM(E, F), we define: N(T) = { e ∈ E : T (E) = 0 } (the null space) R(T) = { f ∈ F : f = T (e) for some e ∈ E } (the range). N(T) is always closed, but R(T) need not be. If ⊥ denotes the operation of taking orthogonal complement, then R(T)^{⊥} = N(T^{∗}). We let FRED(E, F) be the subset of HOM(E, F) consisting of operators invertible modulo com- pact operators: FRED(E, F) = { T ∈ HOM(E, F) : ∃S_{1}, S_{2} ∈ HOM(F, E) so S_{1}T − I ∈ COM(E) and T S_{2} − I ∈ COM(F) }. We note this condition implies S_{1}−S_{2} ∈ COM(F, E) so we can assume S_{1} = S_{2} if we like. The following lemma provides another useful characterization of FRED(E, F): Lemma 1.4.2. The following are equivalent: (a) T ∈ FRED(E, F); (b) T ∈ END(E, F) has dim N(T) < ∞, dim N(T^{∗}) < ∞, R(T) is closed, and R(T^{∗}) is closed. Proof: Let T ∈ FRED(E, F) and let x_{n} ∈ N(T) with |x_{n}| = 1. Then x_{n} = (I − S_{1}T)x_{n} = Cx_{n}. Since C is compact, there exists a convegent

subsequence. This implies the unit sphere in N(T) is compact so N(T) is finite dimensional. Next let y_{n} = T x_{n} and y_{n} → y. We may assume without loss of generality that x_{n} ∈ N(T)^{⊥}. Suppose there exists a constant C so |x_{n}| ≤ C. We have x_{n} = S_{1}y_{n} + (I − S_{1}T)x_{n}. Since S_{1}y_{n} → S_{1}y and since (I − S_{1}T) is compact, we can find a convergent subsequence so x_{n} → x and hence y = lim_{n} y_{n} = lim_{n} T x_{n} = T x is in the range of T so R(T) will be closed. Suppose instead |x_{n}| → ∞. If x_{′n} = x_{n}/|x_{n}| we have T x_{′n} = y_{n}/|x_{n}| → 0. We apply the same argument to find a subsequence x_{′n} → x with T x = 0, |x| = 1, and x ∈ N(T)^{⊥}. Since this is impossible, we conclude R(T) is closed (by passing to a subsequence, one of these two possibilities must hold). Since T^{∗}S_{∗ 1} − I = C_{∗ 1} and S_{∗ 2}T^{∗} − I = C_{∗ 2} we conclude T^{∗} ∈ FRED(F, E) so N(T^{∗}) is finite dimensional and R(T^{∗}) is closed. This proves (a) implies (b). Conversely, suppose N(T) and N(T^{∗}) are finite dimensional and that R(T) is closed. We decompose: E = N(T) ⊕ N(T)^{⊥}  F = N(T^{∗}) ⊕ R(T) where T: N(T)^{⊥} → R(T) is 1-1 and onto. Consequently, we can find a bounded linear operator S so ST = I on N(T)^{⊥} and T S = I on R(T). We extend S to be zero on N(T^{∗}) and compute ST − I = π_{N(T)}  and T S − I = π_{N}(T^{∗}) where π denotes orthogonal projection on the indicated subspace. Since these two projections have finite dimensional range, they are compact which proves T ∈ FRED(E, F). If T ∈ FRED(E, F) we shall say that T is Fredholm. There is a natural law of composition: Lemma 1.4.3. (a) If T ∈ FRED(E, F) then T^{∗} ∈ FRED(F, E). (b) If T_{1} ∈ FRED(E, F) and T_{2} ∈ FRED(F, G) then T_{2}T_{1} ∈ FRED(E, G). Proof: (a) follows from Lemma 1.4.2. If S_{1}T_{1} −I ∈ C(E) and S_{2}T_{2} −I ∈ C(F) then S_{1}S_{2}T_{2}T_{1} − I = S_{1}(S_{2}T_{2} − I)T_{1} + (S_{1}T_{1} − I) ∈ C(E). Similarly T_{2}T_{1}S_{1}S_{2} − I ∈ C(E). If T ∈ FRED(E, F), then we define: index(T) = dim N(T) − dim N(T^{∗}). We note that ISO(E, F) is contained in FRED(E, F) and that index(T) = 0 if T ∈ ISO(E, F).

34 1.4. Fredholm Operators Lemma 1.4.4. (a) index(T) = − index(T^{∗}). (b) If T ∈ FRED(E, F) and S ∈ FRED(F, G) then index(ST) = index(T) + index(S). (c) FRED(E, F) is a open subset of HOM(E, F). (d) index: FRED(E, F) → Z is continuous and locally constant. Proof: (a) is immediate from the definition. We compute N(ST) = N(T) ⊕ T^{−1}(R(T) ∩ N(S)) N(T^{∗}S^{∗}) = N(S^{∗}) ⊕ (S^{∗})^{−1}(R(S^{∗}) ∩ N(T^{∗})) = N(S^{∗}) ⊕ (S^{∗})^{−1}(R(T)^{⊥} ∩ N(S)^{⊥}) so that index(ST) = dim N(T) + dim(R(T) ∩ N(S)) − dim N(S^{∗}) − dim(R(T)^{⊥} ∩ N(S)^{⊥}) = dim N(T) + dim(R(T) ∩ N(S)) + dim(R(T)^{⊥} ∩ N(S)) − dim N(S^{∗}) − dim(R(T)^{⊥} ∩ N(S)^{⊥}) − dim(R(T)^{⊥} ∩ N(S)) = dim N(T) + dim N(S) − dim N(S^{∗}) − dim(R(T)^{⊥}) = dim N(T) + dim N(S) − dim N(S^{∗}) − dim N(T^{∗}) = index(S) + index(T) We prove (c) and (d) as follows. Fix T ∈ FRED(E, F). We decompose: E = N(T) ⊕ N(T)^{⊥} and F = N(T^{∗}) ⊕ R(T) where T: N(T)^{⊥} → R(T) is 1-1 onto. We let π_{1}: E → N(T) be orthog- onal projection. We define E_{1} = N(T^{∗}) ⊕ E and F_{1} = N(T) ⊕ F to be Hilbert spaces by requiring the decompositon to be orthogonal. Let S ∈ HOM(E, F), we define S_{1} ∈ HOM(E_{1}, F_{1}) by: S_{1}(f_{0} ⊕ e) = π_{1}(e) ⊕ (f_{0} + S_{1}(e)). It is clear |S_{1} − S_{′1}| = |S − S′| so the map S → S_{1} defines a continuous map from HOM(E, F) → HOM(E_{1}, F_{1}). Let i_{1}: E → E_{1} be the natural inclusion and let π_{2}: F_{1} → F be the natural projection. Since N(T) and N(T^{∗}) are finite dimensional, these are Fredholm maps. It is immediate from the definition that if S ∈ HOM(E, F) then S = π_{2}S_{1}i_{1}.

If we let T = S and decompose e = e_{0} ⊕e_{1} and f = f_{0} ⊕f_{1} for e_{0} ∈ N(T) and f_{0} ∈ N(T^{∗}), then: T_{1}(f_{0} ⊕ e_{0} ⊕ e_{1}) = e_{0} ⊕ f_{0} ⊕ T e1 so that T_{1} ∈ ISO(E_{1}, F_{1}). Since ISO(E_{1}, F_{1}) is an open subset, there exists ε > 0 so |T −S| < ε implies S_{1} ∈ ISO(E_{1}, F_{1}). This implies S_{1} is Fredholm so S = π_{2}S_{1}i_{1} is Fredholm and FRED(E, F) is open. Furthermore, we can compute index(S) = index(π_{2}) + index(S_{1}) + index(i_{1}) = index(π_{2}) + index(i_{1}) = dim N(T)−dim N(T^{∗}) = index(T) which proves index is locally constant and hence continuous. This completes the proof of the lemma. We present the following example of an operator with index 1. Let ϕn be orthonormal basis for L^{2} as n ∈ Z. Define the one sided shift T ϕ^{n} = { ϕn−1  if n > 0 0  if n = 0 ϕn  if n < 0 then T is surjective so N(T^{∗}) = {0}. Since N(T) is one dimensional, index(T) = 1. Therefore index(T^{n}) = n and index((T^{∗})^{n}) = −n. This proves index: FRED(E, F) → Z is surjective. In the next chapter, we will give several examples of differential operators which have non-zero index. If we specialize to the case E = F then COM(E) is a closed two-sided ideal of END(E) so we can pass to the quotient algebra END(E)/ COM(E). If GL(END(E)/ COM(E)) denotes the group of invertible elements and if π: END(E) → END(E)/ COM(E) is the natural projection, then FRED(E) is π^{−1} of the invertible elements. If C is compact and T Fredholm, T +tC is Fredholm for any t. This implies index(T) = index(T +tC) so we can extend index: GL(END(E)/ COM(E)) → Z as a surjective group homomorphism. Let P: C^{∞}(V) → C^{∞}(W) be a elliptic ΨDO of order d. We construct an elliptic ΨDO S of order −d with S: C^{∞}(W) → C^{∞}(V) so that SP − I and P S − I are infinitely smoothing operators. Then P: H_{s}(V) → H_{d−s}(W)  and S: H_{d−s}(W) → H_{s}(V) are continuous. Since SP − I: H_{s}(V) → H_{t}(V) is continuous for any t, it is compact. Similarly P S − I is a compact operator so both P and S are Fredholm. If f ∈ N(P), then f is smooth by Lemma 1.3.5. Consequently, N(P) and N(P^{∗}) are independent of the choice of s and index(P) is invari- antly defined. Furthermore, if P_{τ} is a smooth 1-parameter family of such operators, then index(P_{τ}) is independent of the parameter τ by Lemma 1.4.4. In particular, index(P) only depends on the homotopy type of the leading symbol of P. In Chapter 3, we will give a topological formula for index(P) in terms of characteristic classes. We summarize our conclusions about index(P) in the following

36 1.4. Fredholm Operators Lemma 1.4.5. Let P: C^{∞}(V) → C^{∞}(W) be an elliptic ΨDO of order d over a compact manifold without boundary. Then: (a) N(P) is a finite dimensional subset of C^{∞}(V). (b) P: H_{s}(V) → H_{s−d}(W) is Fredholm. P has closed range. index(P) does not depend on the particular s chosen. (c) index(P) only depends on the homotopy type of the leading symbol of P within the class of elliptic ΨDO’s of order d.

1.5. Elliptic Complexes, The Hodge Decomposition Theorem, And Poincaré Duality. Let V be a graded vector bundle. V is a collection of vector bundles {V_{j}}_{j∈Z} such that V_{j} ≠ {0} for only a finite number of indices j. We let P be a graded ΨDO of order d. P is a collection of d^{th} order pseudo- differential operators P_{j}: C^{∞}(V_{j}) → C^{∞}(V_{j+1}). We say that (P, V) is a complex if P_{j+1}P_{j} = 0 and σ_{L}P_{j+1}σ_{L}P_{j} = 0 (the condition on the symbol follows from P^{2} = 0 for differential operators). We say that (P, V) is elliptic if:  N(σ_{L}P_{j})(x, ξ) = R(σ_{L}P_{j−1})(x, ξ)  for ξ ≠ 0 or equivalently if the complex is exact on the symbol level. We define the cohomology of this complex by: H^{j}(V, P) = N(P_{j})/ R(P_{j−1}). We shall show later in this section that H^{j}(V, P) is finite dimensional if (P, V) is an elliptic complex. We then define index(P) = _{∑ j}(−1)^{j} dim H^{j}(V, P) as the Euler characteristic of this elliptic complex. Choose a fixed Hermitian inner product on the fibers of V. We use that inner product together with the Riemannian metric on M to define L^{2}(V). We define adjoints with respect to this structure. If (P, V) is an elliptic complex, we construct the associated self-adjoint Laplacian: Δ_{j} = (P^{∗}P)_{j} = P_{∗ j}P_{j} + P_{j−1}P_{∗ j−1}: C^{∞}(V_{j}) → C^{∞}(V_{j}). If p_{j} = σ_{L}(P_{j}), then σ_{L}(Δ_{j}) = p_{∗ j}p_{j} + p_{j−1}p_{∗ j−1}. We can also express the condition of ellipticity in terms of Δ_{j} : Lemma 1.5.1. Let (P, V) be a d^{th} order partial differential complex. Then (P, V) is elliptic if and only if Δ_{j} is an elliptic operator of order 2d for all j. Proof: We suppose that (P, V) is elliptic; we must check σ_{L}(Δ_{j}) is non- singular for ξ ≠ 0. Suppose (p_{∗ j}p_{j} + p_{j−1}p_{∗ j−1})(x, ξ)v = 0. If we dot this equation with v, we see p_{j}v · p_{j}v + p_{∗ j−1}v · p_{∗ j−1}v = 0 so that p_{j}v = p_{∗ j−1}v = 0. Thus v ∈ N(p_{j}) so v ∈ R(p_{j−1}) so we can write v = p_{j−1}w. Since p_{∗ j−1}p_{j−1}w = 0, we dot this equation with w to see p_{j−1}w · p_{j−1}w = 0 so v = p_{j−1}w = 0 which proves Δ_{j} is elliptic. Conversely, let σ_{L}(Δ_{j}) be non- singular for ξ ≠ 0. Since (P, V) is a complex, R(p_{j−1}) is a subset of N(p_{j}).

38 1.5. Elliptic Complexes, Hodge Decomposition Conversely, let v ∈ N(p_{j}). Since σ_{L}(Δ_{j}) is non-singular, we can express v = (p_{∗ j}p_{j} + p_{j−1}p_{∗ j−1})w. We apply p_{j} to conclude p_{j}p_{∗ j}p_{j}w = 0. We dot this equation with p_{j}w to conclude p_{j}p_{∗ j}p_{j}w · p_{j}w = p_{∗ j}p_{j}w · p_{∗ j}p_{j}w = 0 so p_{∗ j}p_{j}w = 0. This implies v = p_{j−1}p_{∗ j−1}w ∈ R(p_{j−1}) which completes the proof. We can now prove the following: Theorem 1.5.2 (Hodge decomposition theorem). Let (P, V) be a d^{th} order ΨDO elliptic complex. Then (a) We can decompose L^{2}(V_{j}) = N(Δ_{j})⊕R(P_{j−1})⊕R(P_{∗ j}) as an orthogonal direct sum. (b) N(Δ_{j}) is a finite dimensional vector space and there is a natural isomor- phism of H^{j}(P, V) ≃ N(Δ_{j}). The elements of N(Δ_{j}) are smooth sections to V_{j}. Proof: We regard Δ_{j}: H_{2d}(V_{j}) → L^{2}(V_{j}). Since this is elliptic, it is Fred- holm. This proves N(Δ_{j}) is finite dimensional. Since Δ_{j} is hypoelliptic, N(Δ_{j}) consists of smooth sections to V. Since Δ_{j} is self adjoint and Fred- holm, R(Δ_{j}) is closed so we may decompose L^{2}(V_{j}) = N(Δ_{j})⊕R(Δ_{j}). It is clear R(Δ_{j}) is contained in the span of R(P_{j−1}) and R(P_{∗ j}). We compute the L^{2} inner product:(P_{∗ j}f, P_{j−1}g) = (f, P_{j}P_{j−1}g) = 0 since P_{j}P_{j−1} = 0. This implies R(P_{j−1}) and R(P_{∗ j}) are orthogonal. Let f ∈ N(Δ_{j}), we take the L^{2} inner product with f to conclude 0 = (Δ_{j}f, f) = (P_{j}f, P_{j}f) + (P_{∗ j−1}f, P_{∗ j−1}f) so N(Δ_{j}) = N(P_{j}) ∩ N(P_{∗ j−1}). This implies R(Δ_{j}) contains the span of R(P_{j−1}) and R(P_{∗ j}). Since these two subspaces are orthogonal and R(Δ_{j}) is closed, R(P_{j−1}) and R(P_{∗ j}) are both closed and we have an orthogonal direct sum: L^{2}(V_{j}) = N(Δ_{j}) ⊕ R(P_{j−1}) ⊕ R(P_{∗ j}). This proves (a). The natural inclusion of N(Δ_{j}) into N(P_{j}) defines a natural map of N(Δ_{j}) → H^{j}(P, V) = N(P_{j})/ R(P_{j−1}). Since R(P_{j−1}) is orthogonal to N(Δ_{j}), this map is injective. If f ∈ C^{∞}(V_{j}) and P_{j}f = 0, we can decom- pose f = f_{0} ⊕ Δ_{j}f_{1} for f_{0} ∈ N(Δ_{j}). Since f and f_{0} are smooth, Δ_{j}f_{1} is smooth so f_{1} ∈ C^{∞}(V_{j}). P_{j}f = P_{j}f_{0} + P_{j}Δ_{j}f_{1} = 0 implies P_{j}Δ_{j}f_{1} = 0. We dot this equation with P_{j}f_{1} to conclude: 0 = (P_{j}f_{1}, P_{j}P_{∗ j}P_{j}f_{1} + P_{j}P_{j−1}P_{∗ j−1}f_{1}) = (P_{∗ j}P_{j}f_{1}, P_{∗ j}P_{j}f_{1})

so P_{∗ j}P_{j}f_{1} = 0 so Δ_{j}f_{1} = P_{j−1}P_{∗ j−1}f_{1} ∈ R(P_{j−1}). This implies f and f0 represent the same element of H^{j}(P, V) so the map N(Δ_{j}) → H^{j}(P, V) is surjective. This completes the proof. To illustrate these concepts, we discuss briefly the de Rham complex. Let T^{∗}M be the cotangent space of M. The exterior algebra Λ(T^{∗}M) is the universal algebra generated by T^{∗}M subject to the relation ξ ∧ ξ = 0 for ξ ∈ T^{∗}M. If {e_{1},…, e_{m}} is a basis for T^{∗}M and if I = {1 ≤ i_{1} < i_{2} < ⋯ < i_{p} ≤ m}, we define e_{I} = e_{i1} ∧ ⋯ ∧ e_{ip}. The {e_{I}} form a basis for Λ(T^{∗}M) which has dimension 2^{m}. If we define |I| = p, then Λ^{p}(T^{∗}M) is the span of the {e_{I}}_{|I|=p}; this is the bundle of p-forms. A section of C^{∞}(Λ^{p}T^{∗}M) is said to be a smooth p-form over M. Let x = (x_{1},…, x_{m}) be local coordinates on M and let {dx_{1},…, dx_{m}} be the corresponding frame for T^{∗}M. If f ∈ C^{∞}(M) = C^{∞}(Λ^{0}(T^{∗}M)), we define:  df = _{∑ k} ∂f ∂xk dx_{k}. If y = (y_{1},…, y_{m}) is another system of local coordinates on M, the iden- tity:  dy_{j} = _{∑ k} ∂yj ∂xk dxk means that d is well defined and is independent of the coordinate system chosen. More generally, we define d(fdx_{I}) = df ∧ dx_{I} so that, for example, d(_{∑ j} f_{j} dx^{j}) = _{∑ j<k} {∂fk ∂xj − ∂fj ∂xk } dx_{j} ∧ dx_{k}. Again this is well defined and independent of the coordinate system. Since mixed partial derivatives commute, d^{2} = 0 so d: C^{∞}(Λ^{p}(T^{∗}M)) → C^{∞}(Λ^{p+1}(T^{∗}M)) forms a complex. Let ξ ∈ T^{∗}M and let ext(ξ): Λ^{p}(T^{∗}M) → Λ^{p+1}(T^{∗}M) be defined by ex- terior multiplication, i.e., ext(ξ)ω = ξ ∧ ω. If we decompose ξ = ∑ ξ_{j} dxj relative to a local coordinate frame, then df = ∑_{j} ∂f/∂x_{j} dx_{j} implies σ(d) = i ext(ξ); the symbol of exterior differentiation is exterior multipli- cation up to a factor of i. Fix ξ ≠ 0 and choose a basis {e_{1},…, e_{m}} for T^{∗}M such that ξ = e_{1}. Then ext(ξ)e^{I} = { 0  if i_{1} = 1 e_{J} for J = {1, i_{1},…, i_{p}}  if i^{1} > 1. From this it is clear that N(ext(ξ)) = R(ext(ξ)) so the de Rham complex is an elliptic complex.

A Riemannian metric on M defines fiber metrics on Λ^{p}(T^{∗}M). If {e_{i}} is an orthonormal local frame for T M we take the dual frame {e_{∗ i}} for T^{∗}M. For notational simplicity, we will simply denote this frame again by {e_{i}}. The corresponding {e_{I}} define an orthonormal local frame for Λ(T^{∗}M). We define interior multiplication int(ξ): Λ^{p}(T^{∗}M) → Λ^{p−1}(T^{∗}M) to be the dual of exterior multiplication. Then: int(e^{1})e^{I} = { e_{j} for J = {i_{2},…, i_{p}}  id i_{1} = 1 0  if i^{1} > 1 so  int(e_{1}) ext(e_{1}) + ext(e_{1}) int(e_{1}) = I. If |ξ|^{2} denotes the length of the covector ξ, then more generally: (i ext(ξ) − i int(ξ))^{2} = |ξ|^{2}I. If δ: C^{∞}(Λ^{p}(T^{∗}M)) → C^{∞}(Λ^{p−1}(T^{∗}M)) is the adjoint of d, then σ_{L}δ = i int(ξ). We let δd + dδ = (d + δ)^{2} = Δ. σ_{L}Δ = |ξ|^{2} is elliptic. The de Rham theorem gives a natural isomorphism from the cohomology of M to H^{∗}(Λ, d):  H^{p}(M; C) = N(d_{p})/ R(d_{p−1}) where we take closed modulo exact forms. The Hodge decomposition the- orem implies these groups are naturally isomorphic to the harmonic p- forms N(Δ_{p}) which are finite dimensional. The Euler-Poincaré character- istic χ(M) is given by: χ(M) = ∑(−1)^{p} dim H^{p}(M; C) = ∑(−1)^{p} dim N(Δ_{p}) = index(d) is therefore the index of an elliptic complex. If M is oriented, we let dvol be the oriented volume element. The Hodge ∗ operator ∗: Λ^{p}(T^{∗}M) → Λ^{m−p}(T^{∗}M) is defined by the identity: ω ∧ ∗ω = (ω · ω) dvol where “·” denotes the inner product defined by the metric. If {e_{i}} is an oriented local frame for T^{∗}M, then dvol = e_{1} ∧ ⋯ ∧ e_{m} and ∗(e_{1} ∧ ⋯ ∧ e_{p}) = e_{p+1} ∧ ⋯ ∧ e_{m}. The following identities are immediate consequences of Stoke’s theorem: ∗∗ = (−1)^{p(m−p)}  and δ = (−1)^{mp+m+1} ∗ d ∗ .

Since Δ = (d+δ)^{2} = dδ+δd we compute ∗Δ = Δ∗ so ∗: N(Δ_{p}) → N(Δ_{m−p}) is an isomorphism. We may regard ∗ as an isomorphism ∗: H^{p}(M; C) → H^{m−p}(M; C); in this description it is Poincaré duality. The exterior algebra is not very suited to computations owing to the large number of signs which enter in the discussion of ∗. When we discuss the signature and spin complexes in the third chapter, we will introduce Clifford algebras which make the discussion of Poincaré duality much easier. It is possible to “roll up” the de Rham complex and define: (d + δ)_{e}: C^{∞}(Λ^{e}(T^{∗}M)) → C^{∞}(Λ^{o}(T^{∗}M)) where Λ^{e}(T^{∗}M) = _{⨁ 2k} Λ^{2k}(T^{∗}M)  and Λ^{o}(T^{∗}M) = _{⨁ 2k+1} Λ^{2k+1} (T^{∗}M) denote the differential forms of even and odd degrees. (d + δ) is an elliptic operator since (d + δ)_{∗ e}(d + δ)_{e} = Δ is elliptic since dim Λ^{e} = dim Λ^{o}. (In this representation (d + δ)_{e} is not self-adjoint since the range and domain are distinct, (d + δ)_{∗ e} = (d + δ)_{o}). It is clear index(d + δ)_{e} = dim N(Δ^{e}) = dim N(Δ^{o}) = χ(M). We can always “roll up” any elliptic complex to form an elliptic complex of the same index with two terms. Of course, the original elliptic complex does not depend upon the choice of a fiber metric to define adjoints so there is some advantage in working with the full complex occasionally as we shall see later. We note finally that if m is even, we can always find a manifold with χ(M) arbitrary so there exist lots of elliptic operators with non-zero index. We shall see that index(P) = 0 if m is odd (and one must consider pseudo- differential operators to get a non-zero index in that case). We summarize these computations for the de Rham complex in the fol- lowing: Lemma 1.5.3. Let Λ(M) = Λ(T^{∗}M) be the complete exterior algebra. (a) d: C^{∞}(Λ^{p}(T^{∗}M)) → C^{∞}(Λ^{p+1}(T^{∗}M)) is an elliptic complex. The sym- bol is σ_{L}(d)(x, ξ) = i ext(ξ). (b) If δ is the adjoint, then σ_{L}(δ)(x, ξ) = −i int(ξ). (c) If Δ_{p} = (dδ + δd)_{p} is the associated Laplacian, then N(Δ_{p}) is finite dimensional and there are natural identifications: N(Δ_{p}) ≃ N(d_{p})/ R(d_{p−1}) ≃ H^{p}(M; C). (d) index(d) = χ(M) is the Euler-Poincaré characteristic of M. (e) If M is oriented, we let ∗ be the Hodge operator. Then ∗∗ = (−1)^{p(m−p)}  and δ = (−1)^{mp+m+1} ∗ d ∗ . Furthermore, ∗: N(Δ_{p}) ≃ N(Δ_{m−p}) gives Poincaré duality. In the next chapter, we will discuss the Gauss-Bonnet theorem which gives a formula for index(d) = χ(M) in terms of curvature.

1.6. The Heat Equation. Before proceeding with our discussion of the index of an elliptic operator, we must discuss spectral theory. We restrict ourselves to the context of a compact self-adjoint operator to avoid unnecessary technical details. Let T ∈ COM(H) be a self-adjoint compact operator on the Hilbert space H. Let  spec(T) = { λ ∈ C : (T − λ) / ∈ GL(H) }. Since GL(H) is open, spec(T) is a closed subset of C. If |λ| > |T |, the series  g(λ) = ^{∞ ∑ n=0} T^{n}/λn+1 converges to define an element of END(H). As (T −λ)g(λ) = g(λ)(T −λ) = −I, λ / ∈ spec(T). This shows spec(T) is bounded. Since T is self-adjoint, N(T − λ̄) = {0} if λ / ∈ R. This implies R(T − λ) is dense in H. Since T x · x is always real, |(T − λ)x · x| ≥ im(λ)| |x|^{2} so that |(T − λ)x| ≥ im(λ)| |x|. This implies R(T − λ) is closed so T − λ is surjective. Since T − λ is injective, λ / ∈ spec(T) if λ ∈ C − R so spec(T) is a subset of R. If λ ∈ [−|T |, |T |], we define E(λ) = { x ∈ H : T x = λx }. Lemma 1.6.1. Let T ∈ COM(H) be self-adjoint. Then dim{E(−|T |) ⊕ E(|T |)} > 0. Proof: If |T | = 0 then T = 0 and the result is clear. Otherwise choose |x_{n}| = 1 so that |T x_{n}| → |T |. We choose a subsequence so T x_{n} → y. Let λ = |T |, we compute: |T^{2}x_{n} − λ^{2}x_{n}|2 = |T^{2}x_{n}|^{2} + |λ^{2}x_{n}|^{2} − 2λ^{2}T^{2}x_{n} · xn ≤ 2λ^{4} − 2λ^{2}|T x_{n}|^{2} → 0. Since T x_{n} → y, T^{2}x_{n} → T y. Thus λ^{2}x_{n} → T y. Since λ^{2} ≠ 0, this implies x_{n} → x for x = T y/λ^{2}. Furthermore |T^{2}x − λ^{2}x| = 0. Since (T^{2} − λ^{2}) = (T − λ)(T + λ), either (T + λ)x = 0 so x ∈ E(−λ) ≠ {0} or (T − λ)(y_{1}) = 0 for y_{1} = (T + λ)x ≠ 0 so E(λ) ≠ {0}. This completes the proof. If λ ≠ 0, the equation T x = λx implies the unit disk in E(λ) is compact and hence E(λ) is finite dimensional. E(λ) is T -invariant. Since T is self- adjoint, the orthogonal complement E(λ) is also T -invariant. We take an orthogonal decomposition H = E(|T |) ⊕ E(−|T |) ⊕ H_{1}. T respects this decomposition; we let T_{1} be the restriction of T to H_{1}. Clearly |T_{1}| ≤ |T |. If we have equality, then Lemma 1.6.1 implies there exists x ≠ 0 in H1

The Heat Equation 43

so T_{1}x = ±|T |x, which would be false. Thus |T_{1}| < |T |. We proceed inductively to decompose: H = E(λ_{1}) ⊕ E(−λ_{1}) ⊕ ⋯ ⊕ E(λ_{n}) ⊕ E(−λ_{n}) ⊕ Hn where λ_{1} > λ_{2} > ⋯ > λ_{n} and where |T_{n}| < λ_{n}. We also assume E(λ_{n}) ⊕ E(−λ_{n}) ≠ {0}. We suppose that the λ_{n} do not converge to zero but converge to ε positive. For each n, we choose x_{n} so |x_{n}| = 1 and T x_{n} = ±λ_{n}x_{n}. Since T is self-adjoint, this is an orthogonal decomposition; |x_{j} − x_{k}| = √2. Since T is compact, we can choose a convergent subsequence λ_{n}x_{n} → y. Since λ_{n} → ε positive, this implies x_{n} → x. This is impossible so therefore the λ_{n} → 0. We define H_{0} = ⋂_{n} H_{n} as a closed subset of H. Since |T | < λ_{n} for all n on H_{0}, |T | = 0 so T = 0 and H_{0} = E(0). This defines a direct sum decomposition of the form: H = _{⨁ k} E(μ_{k}) ⊕ E(0) where the μ_{k} ∈ R are the non-zero subspace of the E(λ_{n}) and E(−λ_{n}). We construct a complete orthonormal system {ϕ_{n}}_{∞ n=1} for H with either ϕ_{n} ∈ E(0) or ϕ_{n} ∈ E(μ_{k}) for some k. Then T ϕ_{n} = λ_{n}ϕ_{n} so ϕ_{n} is an eigenvector of T. This proves the spectral decomposition for self-adjoint compact operators: Lemma 1.6.2. Let T ∈ COM(H) be self-adjoint. We can find a complete orthonormal system for H consisting of eigenvectors of T. We remark that this need not be true if T is self-adjoint but not compact or if T is compact but not self-adjoint. We can use this lemma to prove the following: Lemma 1.6.3. Let P: C^{∞}(V) → C^{∞}(V) be an elliptic self-adjoint ΨDO of order d > 0. (a) We can find a complete orthonormal basis {ϕ_{n}}_{∞ n=1} for L^{2}(V) of eigen- vectors of P. P ϕ_{n} = λ_{n}ϕ_{n}. (b) The eigenvectors ϕ_{n} are smooth and lim_{n→∞} |λ_{n}| = ∞. (c) If we order the eigenvalues |λ_{1}| ≤ |λ_{2}| ≤ ⋯ then there exists a constant C > 0 and an exponent δ > 0 such that |λ_{n}| ≥ Cn^{δ} if n > n_{0} is large. Remark: The estimate (c) can be improved to show |λ_{n}| ∼ n^{d/m} but the weaker estimate will suffice and is easier to prove. Proof: P: H_{d}(V) → L^{2}(V) is Fredholm. P: N(P)^{⊥} ∩ H_{d}(M) → N(P)^{⊥} ∩ L^{2}(V) is 1-1 and onto; by Gärding’s inequality P ϕ ∈ L^{2} implies ϕ ∈ H_{d}. We let S denote the inverse of this map and extend S to be zero on the finite dimensional space N(P). Since the inclusion of H_{d}(M) into L^{2}(V) is compact, S is a compact self-adjoint operator. S is often referred to

as the Greens operator. We find a complete orthonormal system {ϕ_{n}} of eigenvectors of S. If Sϕ_{n} = 0 then P ϕ_{n} = 0 since N(S) = N(P). If Sϕ_{n} = μ_{n}ϕ_{n} ≠ 0 then P ϕ_{n} = λ_{n}ϕ_{n} for λ_{n} = μ_{−1 n}. Since the μ_{n} → 0, the |λ_{n}| → ∞. If k is an integer such that dk > 1, then P^{k} − λ_{k n} is elliptic. Since (P^{k} − λ_{k n})ϕ_{n} = 0, this implies ϕ_{n} ∈ C^{∞}(V). This completes the proof of (a) and (b). By replacing P with P^{k} we replace λ_{n} by λ_{k n}. Since kd > _{m 2} if k is large, we may assume without loss of generality that d > _{m 2} in the proof of (c). We define:  |f|_{∞,0} = _{sup x∈M} |f(x)|  for f ∈ C^{∞}(V). We estimate:  |f|_{∞,0} ≤ C|f|_{d} ≤ C(|P f|_{0} + |f|_{0}). Let F (a) be the space spanned by the ϕ_{j} where |λ_{j}| ≤ a and let n = dim F (a). We estimate n = n(a) as follows. We have |f|_{∞,0} ≤ C(1 + a)|f|_{0}  on F (a). Suppose first V = M × C is the trivial line bundle. Let c_{j} be complex constants, then this estimate shows: ||||_{n ∑ j=1} c_{j}ϕ_{j}(x)|||| ≤ C(1 + a){_{n ∑ j=1} |c_{j}|^{2}}^{1/2}. If we take c_{j} = ϕ̄j(x) then this yields the estimate:_{n ∑ j=1} ϕ_{j}(x)ϕ̄j(x) ≤ C(1 + a){_{n ∑ j=1} ϕ_{j}(x)ϕ̄j(x)}^{1/2}, i.e.,  _{n ∑ j=1} ϕ_{j}(x)ϕ̄j(x) ≤ C^{2}(1 + a)^{2}. We integrate this estimate over M to conclude n ≤ C^{2}(1 + a)^{2} vol(M) or equivalently  C_{1}(n − C_{2})^{1/2} ≤ a = |λ_{n}| from which the desired estimate follows.

If dim V = k, we choose a local orthonormal frame for V to decompose ϕ_{j} into components ϕ_{u j} for 1 ≤ u ≤ k. We estimate: ||||_{n ∑ j=1} c_{u j}ϕ_{u j}(x)|||| ≤ C(1 + a) {_{n ∑ j=1} |c_{u j}|^{2}}^{1/2}  for u = 1,…, k. If we let c_{u j} = ϕ̄^{u} j (x), then summing over u yields: ||||^{n ∑ j=1} ϕ^{j}ϕ̄j(x)|||| ≤ kC(1 + a) ||||^{n ∑ j=1} ϕ^{j}ϕ̄j(x)||||1/2 since each term on the left hand side of the previous inequality can be estimated seperately by the right hand side of this inequality. This means that the constants which arise in the estimate of (c) depend on k, but this causes no difficulty. This completes the proof of (c). The argument given above for (c) was shown to us by Prof. B. Allard (Duke University) and it is a clever argument to avoid the use of the Schwarz kernel theorem. Let P: C^{∞}(V) → C^{∞}(V) be an elliptic ΨDO of order d > 0 which is self-adjoint. We say P has positive definite leading symbol if there exists p(x, ξ): T^{∗}M → END(V) such that p(x, ξ) is a positive definite Hermitian matrix for ξ ≠ 0 and such that σP − p ∈ S^{d−1} in any coordinate system. The spectrum of such a P is not necessarily non-negative, but it is bounded from below as we shall show. We construct Q_{0} with leading symbol √p and let Q = Q_{∗ 0}Q_{0}. Then by hypothesis P − Q ∈ S^{d−1}. We compute: (P f, f) = ((P − Q)f, f) + (Qf, f) |((P − Q)f, f)| ≤ C|f|_{d/2}|(P − Q)f|_{d/2} ≤ C|f|_{d/2}|f|d/2−1 (Qf, f) = (Q_{0}f, Q_{0}f)  and |f|_{2 d/2} < C|f|_{2 0} + C|Q_{0}f|_{2 0}. We use this to estimate for any ε > 0 that: |((P − Q)f, f)| ≤ C|f|_{d/2}(|f|_{d/2−1}) ≤ ε|f|_{2 d/2} + C(ε)|f|_{d/2}|f|0 ≤ 2ε|f|_{2 d/2} + C(ε)|f|^{2} 0 ≤ 2Cε|Q_{0}f|_{2 0} + C(ε)|f|_{2 0}. Choose ε so 2Cε ≤ 1 and estimate: (P f, f) ≥ (Q_{0}f, Q_{0}f) − |((P − Q)f, f)| ≥ −C(ε)|f|_{2 0}. This implies:

Lemma 1.6.4. Let P: C^∞(V) → C^∞(V) be an elliptic ΨDO of order d > 0 which is self-adjoint with positive definite leading symbol. Then spec(P) is contained in [−C, ∞) for some constant C.

We fix such a P henceforth. The heat equation is the partial differential equation: (

d / (dt) + P ) f(x, t) = 0 for t ≥ 0 with f(x, 0) = f(x).

At least formally, this has the solution f(x, t) = e^{−tP} f(x). We decompose f(x) = ∑ c_nϕ_n(x) for c_n = (f, ϕ_n) in a generalized Fourier series. The solution of the heat equation is given by:

f(x, t) = ∑

n

e^{−tλn} c_nϕ_n(x).

Proceeding formally, we define:

K(t, x, y) = ∑

n

e^{−tλn} ϕ_n(x) ⊗ ϕ̄_n(y): V_y → V_x

so that:

e^{−tP} f(x) = ∫_M K(t, x, y)f(y) dvol(y)

= ∑

n

e^{−tλn} ϕ_n(x) ∫_M f(y) · ϕ_n(y) dvol(y).

We regard K(t, x, y) as an endomorphism from the fiber of V over y to the fiber of V over x.

We justify this purely formal procedure using Lemma 1.3.4. We estimate that:

|ϕ_n|_{∞,k} ≤ C(|ϕ_n|_0 + |P^jϕ|_0) = C(1 + |λ_n|^j) where jd > k + m/2.

Only a finite number of eigenvalues of P are negative by Lemma 1.6.4. These will not affect convergence questions, so we may assume λ > 0. Estimate:

e^{−tλ}λ^j ≤ t^{−j}C(j)e^{−tλ/2}

to compute:

|K(t, x, y)|_{∞,k} ≤ t^{−j(k)}C(k) ∑

n

e^{−tλn/2}.

Since |λ| ≥ Cn^δ for δ > 0 and n large, the series can be bounded by ∑

n>0

e^{−tnδ/2}

which converges. This shows K(t, x, y) is an infinitely smooth function of (t, x, y) for t > 0 and justifies all the formal procedures involved. We compute Tr_{L2} e^{−tP} = ∑ e^{−tλn} = ∫_{M} Tr_{Vx} K(t, x, x) dvol(x). We can use this formula to compute the index of Q: Lemma 1.6.5. Let Q: C^{∞}(V) → C^{∞}(W) be an elliptic ΨDO of order d > 0. Then for t > 0, e^{−tQ^{∗}Q} and e^{−tQQ∗} are in Ψ_{−∞} with smooth kernel functions and index(Q) = Tr e^{−tQ^{∗}Q} − Tr e^{−tQQ∗}  for any t > 0. Proof: Since e^{−tQ^{∗}Q} and e^{−tQQ∗} have smooth kernel functions, they are in Ψ_{−∞} so we must only prove the identity on index(Q). We define E_{0}(λ) = { ϕ ∈ L^{2}(V) : Q^{∗}Qϕ = λϕ } and E_{1}(λ) = { ϕ ∈ L^{2}(W) : QQ^{∗}ϕ = λϕ }. These are finite dimensional subspaces of smooth sections to V and W. Because Q(Q^{∗}Q) = (QQ^{∗})Q and Q^{∗}(QQ^{∗}) = (Q^{∗}Q)Q^{∗}, Q and Q^{∗} define maps:  Q: E_{0}(λ) → E_{1}(λ)  and Q^{∗}: E_{1}(λ) → E_{0}(λ). If λ ≠ 0, λ = Q^{∗}Q: E_{0}(λ) → E_{1}(λ) → E_{0}(λ) is an isomorphism so dim E_{0}(λ) = dim E_{1}(λ). We compute: Tr e^{−tQ^{∗}Q} − Tr e−tQQ∗ = _{∑ λ} e^{−tλ}{dim E_{0}(λ) − dim E_{1}(λ)} = e^{−t0}{dim E_{0}(0) − dim E_{1}(0)} = index(Q) which completes the proof. If (Q, V) is a elliptic complex, we use the same reasoning to conclude e^{−tΔi} is in Ψ_{−∞} with a smooth kernel function. We define E_{i}(λ) = { ϕ ∈ L^{2}(V_{i}) : Δ_{i}ϕ = λϕ }. Then Q_{i}: E_{i}(λ) → E_{i+1}(λ) defines an acyclic complex (i.e., N(Q_{i}) = R(Q_{i−1})) if λ ≠ 0 so that ∑(−1)^{i} dim E_{i}(λ) = 0 for λ ≠ 0. Therefore  index(Q) = _{∑ i}(−1)^{i} Tr(e^{−tΔi}) and Lemma 1.6.5 generalizes to the case of elliptic complexes which are not just two term complexes. Let P be an elliptic ΨDO of order d > 0 which is self-adjoint with positive definite leading symbol. Then spec(P) is contained in [−C, ∞) for some constant C. For λ / ∈ spec(P), (P − λ)^{−1} ∈ END(L^{2}(V)) satisfies: |(P − λ)^{−1}| = dist(λ, spec(P))^{−1} = _{inf n} |λ − λ^{n}|^{−1}.

[figure: x-axis; spectrum of P; y-axis; R]

The function |(P − λ)^{−1}| is a continuous function of λ ∈ C − spec(P). We use the Riemann integral with values in the Banach space END(L^{2}(V)) to define:  e^{−tP} = 1 2πi ∫_{γ} e^{−tλ}(P − λ)^{−1} dλ where γ is the path about [−C, ∞) given by the union of the appropriate pieces of the straight lines Re(λ+C +1) = ± Im(λ) pictured below. We let R be the closed region of C consisting of γ together with that component of C − γ which does not contain [−C, ∞). We wish to extend (P − λ)^{−1} to H_{s}. We note that |λ − μ|^{−1} ≤ C  for λ ∈ R, μ ∈ spec(P) so that |(P − λ)^{−1}f|_{0} < C|f|_{0}. We use Lemma 1.3.5 to estimate: |(P − λ)^{−1}f|kd ≤ C{|P^{k}(P − λ)^{−1}f|_{0} + |(P − λ)^{−1}f|_{0}} ≤ C{|P^{k−1}f|_{0} + |λP^{k−1}(P − λ)^{−1}f|_{0} + |f|_{0}} ≤ C{|f|_{kd−d} + |λ| |(P − λ)^{−1}f|_{kd−d}}. If k = 1, this implies |(P − λ)^{−1}f|_{d} ≤ C(1 + |λ|)|f|_{0}. We now argue by induction to estimate: |(P − λ)^{−1}f|_{kd} ≤ C(1 + |λ|)^{k−1}|f|_{kd−d}. We now interpolate. If s > 0, choose k so kd ≥ s > kd − d and estimate: |(P − λ)^{−1}f|s ≤ C|(P − λ)^{−1}f|_{kd} ≤ C(1 + |λ|)^{k−1}|f|kd−d ≤ C(1 + |λ|)^{k−1}|f|_{s}.

The Heat Equation 49

Similarly, if s < 0, we use duality to estimate: |((P − λ)^{−1}f, g)| = |(f, (P − λ̄)^{−1}g)| ≤ C(1 + |λ|)^{k−1}|f|_{s}|g|−s which by Lemma 1.3.4 shows |(P − λ)^{−1}f|_{s} < C(1 + |λ|)^{k−1}|f|s in this case as well. Lemma 1.6.6. Let P be a elliptic ΨDO of order d > 0 which is self- adjoint. We suppose the leading symbol of P is positive definite. Then: (a) Given s there exists k = k(s) and C = C(s) so that |(P − λ)^{−1}f|_{s} ≤ C(1 + |λ|)^{k}|f|s for all λ ∈ R. (b) Given j there exists k = k(j, d) so (P^{k} − λ)^{−1} represents a smoothing operator with C^{j} kernel function which is of trace class for any λ ∈ R. Proof: (a) follows from the estimates previously. To prove (b) we let K_{k}(x, y, λ) = _{∑ n} 1 λ_{k n} − λϕ_{n}(x) ⊗ ϕ_{n}(y) be the kernel of (P^{k} − λ)^{−1}. The region R was chosen so |λ̃ − λ| ≥ ε|λ̃| for some ε > 0 and λ̃ ∈ R^{+}. Therefore |(λ^{k n} − λ)|^{−1} ≤ ε|λ^{k n}|^{−1} ≤ ε^{−1}n−kδ by Lemma 1.6.3. The convergence of the sum defining K_{k} then follows using the same arguments as given in the proof that e^{−tP} is a smoothing operator. This technical lemma will be used in the next subsection to estimate various error terms which occur in the construction of a parametrix.

1.7. Local Formula for The Index of an Elliptic Operator. In this subsection, let P: C^{∞}(V) → C^{∞}(V) be a self-adjoint elliptic partial differential operator of order d > 0. Decompose the symbol σ(P) = p_{d} + ⋯ + p_{0} into homogeneous polymonials p_{j} of order j in ξ ∈ T^{∗}M. We assume p_{d} is a positive definite Hermitian matrix for ξ ≠ 0. Let the curve γ and the region R be as defined previously. The operator (P − λ)^{−1} for λ ∈ R is not a pseudo-differential operator. We will approximate (P − λ)^{−1} by a pseudo-differential operator R(λ) and then use that approximation to obtain properties of exp(−tP). Let U be an open subset of R^{m} with compact closure. Fix d ∈ Z. We make the following definition to generalize that given in section 1.2: Definition. q(x, ξ, λ) ∈ S^{k}(λ)(U) is a symbol of order k depending on the complex parameter λ ∈ R if (a) q(x, ξ, λ) is smooth in (x, ξ, λ) ∈ R^{m} ×R^{m} ×R, has compact x-support in U and is holomorphic in λ. (b) For all (α, β, γ) there exist constants C_{α,β,γ} such that: |D_{α x}D_{β ξ}D_{γ λ}q(x, ξ, γ)| ≤ C_{α,β,γ} (1 + |ξ| + |λ|^{1/d})^{k−|β|−d|γ|} . We say that q(x, ξ, γ) is homogeneous of order k in (ξ, λ) if q(x, tξ, t^{d}λ) = t^{k}q(x, ξ, γ) for t ≥ 1. We think of the parameter λ as being of order d. It is clear if q is ho- mogeneous in (ξ, λ), then it satisfies the decay conditions (b). Since the p_{j} are polynomials in ξ, they are regular at ξ = 0 and define elements of S^{j}(λ). If the p_{j} were only pseudo-differential, we would have to smooth out the singularity at ξ = 0 which would destroy the homogeneity in (ξ, λ) and they would not belong to S^{j}(λ); equivalently, d_{α ξ}p_{j} will not exhibit decay in λ for |α| > j if p_{j} is not a polynomial. Thus the restriction to differential operators is an essential one, although it is possible to discuss some results in the pseudo-differential case by taking more care with the estimates involved. We also note that (p_{d} − λ)^{−1} ∈ S^{−d}(λ) and that the spaces S^{∗}(λ) form a symbol class closed under differentiation and multiplication. They are suitable generalizations of ordinary pseudo-differential operators discussed earlier. We let Ψ_{k}(λ)(U) be the set of all operators Q(λ): C_{∞ 0}(U) → C_{∞ 0}(U) with symbols q(x, ξ, γ) in S^{k}(λ) having x-support in U; we let q = σQ. For any fixed λ, Q(λ) ∈ Ψ_{k}(U) is an ordinary pseudo-differential operator of order k. The new features arise from the dependence on the parameter λ. We extend the definition of ∼ given earlier to this wider class by: q ∼ _{∑ j} qj

if for every k > 0 there exists n(k) so n ≥ n(k) implies q − ∑_{j≤n} q_{j} ∈ S^{−k}(λ). Lemmas 1.2.1, 1.2.3(b), and 1.3.2 generalize easily to yield the following Lemma. We omit the proofs in the interests of brevity as they are identical to those previously given with only minor technical modifications. Lemma 1.7.1. (a) Let Q_{i} ∈ Ψ_{ki}(λ)(U) with symbol q_{i}. Then Q_{1}Q_{2} ∈ Ψ_{k_{1}+k2} (λ)(U) has symbol q where:  q ∼ _{∑ α} d_{α ξ}q_{1}D_{α x}q_{2}/α! . (b) Given n > 0 there exists k(n) > 0 such that Q ∈ Ψ_{−k(n)} (λ)(U) implies Q(λ) defines a continuous map from H_{−n} → H_{n} and we can estimate the operator norm by:  |Q(λ)|_{−n,n} ≤ C(1 + |λ|)^{−n}. (c) If h: U → U is a diffeomorphism, then h_{∗}: Ψ_{k}(λ)(U) → Ψ_{k}(λ)(U) and σ(h_{∗}P) − p(h^{−1}x_{1}, (dh^{−1}(x_{1}))^{t}ξ_{1}, λ) ∈ S^{k−1}(λ)(U). As before, we let Ψ(λ)(U) = ⋃_{n} Ψ_{n}(λ)(U) be the set of all pseudo- differential operators depending on a complex parameter λ ∈ R defined over U. This class depends on the order d chosen and also on the region R, but we supress this dependence in the interests of notational clarity. There is no analogue of the completeness of Lemma 1.2.8 for such symbols since we require analyticity in λ. Thus in constructing an approximation to the parametrix, we will always restrict to a finite sum rather than an infinite sum. Using (c), we extend the class Ψ(λ) to compact manifolds using a par- tition of unity argument. (a) and (b) generalize suitably. We now turn to the question of ellipticity. We wish to solve the equation: σ(R(λ)(P − λ)) − I ∼ 0. Inductively we define R(λ) with symbol r_{0} + r_{1} + ⋯ where r_{j} ∈ S^{−d−j} (λ). We define p_{′j}(x, ξ, λ) = p_{j}(x, ξ) for j < d and p_{′d}(x, ξ, λ) = p_{d}(x, ξ) − λ. Then σ(P − λ) = ∑_{d j=0} p_{′j}. We note that p_{′j} ∈ S^{j}(λ) and that p′d^{−1} ∈ S^{−d}(λ) so that (P − λ) is elliptic in a suitable sense. The essential feature of this construction is that the parameter λ is absorbed in the leading symbol and not treated as a lower order perturbation. The equation σ(R(λ)(P − λ)) ∼ I yields:_{∑ α,j,k} d^{α ξ}r^{j} · D^{α x}p^{′k}/α! ∼ I.

52 1.7. Local Formula for the Index We decompose this series into terms homogeneous of order −n to write:_{∑ n}_{∑ |α|+j+d−k=n} d_{α ξ}r_{j} · D_{α x}p_{′k}/α! ∼ I where j, k ≥ 0 and k ≤ d. There are no terms with n < 0, i.e., positive order. If we investigate the term with n = 0, we arrive at the condition r_{0}p_{′d} = I so r_{0} = (p_{d} − λ)^{−1} and inductively: r_{n} = −r_{0} _{∑ |α|+j+d−k=n j<n} d_{α ξ}r_{j}D_{α x}p_{′k}/α! . If k = d then |α| > 0 so D_{α x}p_{′k} = D_{α x}p_{k} in this sum. Therefore we may replace p_{′k} by p_{k} and write r_{n} = −r_{0} _{∑ |α|+j+d−k=n j<n} d_{α ξ}r_{j}D_{α x}p_{k}/α! . In a similar fashion, we can define R(λ) so σ((P − λ)R(λ) − I) ∼ 0. This implies σ(R(λ)−R(λ)) ∼ 0 so R(λ) provides a formal left and right inverse. Since such an inverse is unique modulo lower order terms, R(λ) is well defined and unique modulo lower order terms in any coordinate system. We define R(λ) globally on M using a partition of unity argument. To avoid convergence questions, we shall let R(λ) have symbol r^{0} + ⋯ + rn0 where n_{0} is chosen to be very large. R(λ) is unique modulo Ψ_{−n_{0}−d}(λ). For notational convenience, we supress the dependence of R(λ) upon n_{0}. R(λ) gives a good approximation to (P − λ)^{−1} in the following sense: Lemma 1.7.2. Let k > 0 be given. We can choose n_{0} = n_{0}(k) so that |{(P − λ)^{−1} − R(λ)}f|_{k} ≤ C_{k}(1 + |λ|)^{−k}|f|_{−k} for λ ∈ R, f ∈ C^{∞}(V). Thus (P − λ)^{−1} is approximated arbitrarily well by the parametrix R(λ) in the operator norms as λ → ∞. Proof: We compute: |{(P − λ)^{−1} − R(λ)}f|k = |(P − λ)^{−1}{I − (P − λ)R(λ)}f|k ≤ C_{k}(1 + |λ|^{ν})|(I − (P − λ)R(λ))f|k by Lemma 1.6.6. Since I − (P − λ)R(λ) ∈ S^{−n0} , we use Lemma 1.7.1 to complete the proof. We define E(t) = _{1 2πi} ∫_{γ} e^{−tλ}R(λ) dλ. We will show shortly that this has a smooth kernel K′(t, x, y). Let K(t, x, y) be the smooth kernel of e^{−tP}. We

will use Lemma 1.2.9 to estimate the difference between these two kernels. We compute:E(t) − e^{−tP} = 1 2πi ∫_{γ} e^{−tλ}(R(λ) − (P − λ)^{−1}) dλ. We assume 0 < t < 1 and make a change of variables to replace tλ by λ. We use Cauchy’s theorem to shift the resulting path tγ inside R back to the original path γ where we have uniform estimates. This expresses: E(t) − e^{−tP} = 1 2πi ∫_{γ} e^{−λ}(R(t^{−1}λ) − (P − t^{−1}λ))t^{−1} dλ We estimate therefore: |E(t) − e^{−tP} |−k,k ≤ C_{k} ∫_{γ} |e^{−λ}(1 + t^{−1}|λ|)^{−k} dλ| ≤ C^{k}tk provided n_{0} is large enough. Lemma 1.2.9 implies Lemma 1.7.3. Let k be given. If n_{0} is large enough, we can estimate: |K(t, x, y) − K′(t, x, y)|_{∞,k} ≤ C_{k}t^{k}  for 0 < t < 1. This implies that K′ approximates K to arbitrarily high jets as t → 0. We now study the operator E(t). We define e_{n}(t, x, ξ) = 1 2πi ∫_{γ} e^{−tλ}r_{n}(x, ξ, λ) dλ, then E(t) is a ΨDO with symbol e_{0} + ⋯ + e_{n0} . If we integrate by parts in λ we see that we can express e_{n}(t, x, ξ) = 1 2πi 1 tk ∫_{γ} e^{−tλ} dk dλk r_{n}(x, ξ, λ) dλ. Since dk dλk rn is homogeneous of degree −d − n − kd in (ξ, λ), we see that e_{n}(t, x, ξ) ∈ S^{−∞} for any t > 0. Therefore we can apply Lemma 1.2.5 to conclude E_{n}(t) which has symbol e_{n}(t) is represented by a kernel function defined by:  K^{n}(t, x, y) = ∫ e^{i(x−y)·ξ} e^{n}(t, x, ξ) dξ.

We compute: K_{n}(t, x, x) = 1 2πi ∫∫_{γ} e^{−tλ}r_{n}(x, ξ, λ) dλ dξ. We make a change of variables to replace λ by t^{−1}λ and ξ by t^{−1/d}ξ to compute: K_{n}(t, x, x) = t^{−_{^{m} d} −1} 1 2πi ∫∫_{γ} e^{−λ}r_{n}(x, t^{^{−1} d} ξ, t^{−1}λ) dλ dξ = t^{−_{^{m} d} −1+^{n+d} d} 1 2πi ∫∫_{γ} e^{−λ}r_{n}(x, ξ, λ) dλ dξ = t^{^{n−m} d} e_{n}(x) where we let this integral define e_{n}(x). Since p_{d}(x, ξ) is assumed to be positive definite, d must be even since pd is a polynomial in ξ. Inductively we express r_{n} as a sum of terms of the form:  r_{j1 0}q_{1}r_{j2 0}q_{2} … r^{jk} 0 where the q_{k} are polynomials in (x, ξ). The sum of the degrees of the q_{k} is odd if n is odd and therefore r_{n}(x, −ξ, λ) = −r_{n}(x, ξ, λ). If we replace ξ by −ξ in the integral defining e_{n}(x), we conclude e_{n}(x) = 0 if n is odd. Lemma 1.7.4. Let P be a self-adjoint elliptic partial differential operator of order d > 0 such that the leading symbol of P is positive definite for ξ ≠ 0. Then: (a) If we choose a coordinate system for M near a point x ∈ M and choose a local frame for V, we can define e_{n}(x) using the complicated combinatorial recipe given above. e_{n}(x) depends functorially on a finite number of jets of the symbol p(x, ξ). (b) If K(t, x, y) is the kernel of e^{−tP} then K(t, x, x) ∼ ^{∞ ∑ n=0} t^{^{n−m} d} e^{n}(x)  as t → 0+ i.e., given any integer k there exists n(k) such that: ||||K(t, x, x) −_{∑ n≤n(k)} t^{^{n−m} d} e_{n}(x)||||_{∞,k} < C_{k}t^{k}  for 0 < t < 1. (c) e_{n}(x) ∈ END(V, V) is invariantly defined independent of the coordinate system and local frame for V.

Of an Elliptic Operator 55

(d) e_{n}(x) = 0 if n is odd. Proof: (a) is immediate. We computed that K′(t, x, x) = _{n0 ∑ n=0} t^{^{n−m} d} e_{n}(x) so (b) follows from Lemma 1.7.3. Since K(t, x, x) does not depend on the coordinate system chosen, (c) follows from (b). We computed (d) earlier to complete the proof. We remark that this asymptotic representation of K(t, x, x) exists for a much wider class of operators P. We refer to the literature for further details. We shall give explicit formulas for e_{0}, e_{2}, and e_{4} in section 4.8 for certain examples arising in geometry. The invariants e_{n}(x) = e_{n}(x, P) are sections to the bundle of endomor- phisms, END(V). They have a number of functorial properties. We suma- rize some of these below. Lemma 1.7.5. (a) Let P_{i}: C^{∞}(V_{i}) → C^{∞}(V_{i}) be elliptic self-adjoint partial differential operators of order d > 0 with positive definite leading symbol. We form P = P_{1} ⊕ P_{2}: C^{∞}(V_{1} ⊕ V_{2}) → C^{∞}(V_{1} ⊕ V_{2}). Then P is an elliptic self- adjoint partial differential operator of order d > 0 with positive definite leading symbol and e_{n}(x, P_{1} ⊕ P_{2}) = e_{n}(x, P_{1}) ⊕ e_{n}(x, P_{2}). (b) Let P_{i}: C^{∞}(V_{i}) → C^{∞}(V_{i}) be elliptic self-adjoint partial differential operators of order d > 0 with positive definite leading symbol defined over different manifolds M_{i}. We let P = P_{1} ⊗ 1 + 1 ⊗ P_{2}: C^{∞}(V_{1} ⊗ V_{2}) → C^{∞}(V_{1} ⊗ V_{2}) over M = M_{1} × M_{2}. Then P is an elliptic self-adjoint partial differential operator of order d > 0 with positive definite leading symbol over M and e_{n}(x, P) = _{∑ p+q=n} e_{p}(x_{1}, P_{1}) ⊗ e_{q}(x_{2}, P_{2}). (c) Let P: C^{∞}(V) → C^{∞}(V) be an elliptic self-adjoint partial differential operator of order d > 0 with positive definite leading symbol. We decom- pose the total symbol of P in the form: p(x, ξ) = _{∑ |α|≤d} p_{α}(x)ξ^{α}. Fix a local frame for V and a system of local coordinates on M. We let indices a, b index the local frame and let p_{α} = p_{abα} give the components 56 1.7. Local Formula for the Index of the matrix p_{α}. We introduce formal variables p_{abα/β} = D_{β x}p_{abα} for the jets of the symbol of P. Define ord(p_{abα/β}) = |β| + d − |α|. Then e_{n}(x, P) can be expressed as a sum of monomials in the {p_{abα/β}} variables which are homogeneous of order n in the jets of the symbol as discussed above and with coefficients which depend smoothly on the leading symbol {p_{abα}}_{|α|=d}. (d) If the leading symbol of P is scalar, then the invariance theory can be simplified. We do not need to introduce the components of the symbol ex- plicitly and can compute e_{n}(x, P) as a non-commutative polynomial which is homogeneous of order n in the {p_{α/β}} variables with coefficients which depend smoothly upon the leading symbol. Remark: The statement of this lemma is somewhat technical. It will be convenient, however, to have these functorial properties precisely stated for later reference. This result suffices to study the index theorem. We shall give one more result which generalizes this one at the end of this section useful in studying the eta invariant. The objects we are studying have a bigrading; one grading comes from counting the number of derivatives in the jets of the symbol, while the other measures the degree of homogeneity in the (ξ, λ) variables. Proof: (a) and (b) follow from the identities:^{e−t(P^{1}⊕P^{2})} = e^{−tP1} ⊕ e−tP2^{e−t(P^{1}⊗1+1⊗P^{2})} = e^{−tP1} ⊗ e−tP2 so the kernels satisfy the identities: K(t, x, x, P_{1} ⊕ P_{2}) = K(t, x, x, P_{1}) ⊕ K(t, x, x, P_{2}) K(t, x, x, P_{1} ⊗ 1 + 1 ⊗ P_{2}) = K(t, x_{1}, x_{1}, P_{1}) ⊗ K(t, x_{2}, x_{2}, P_{2}). We equate equal powers of t in the asymptotic series: ∑ t^{^{n−m} d} e_{n}(x, P_{1} ⊕ P_{2}) ∼ ∑ t^{^{n−m} d} e_{n}(x, P_{1}) ⊕ ∑ t^{^{n−m} d} e_{n}(x, P_{2}) ∑ t^{^{n−m} d} e_{n}(x, P_{1} ⊗ 1 + 1 ⊗ P_{2}) ∼ {∑ t^{^{p−m1} d} e^{p}(x^{1}, P^{1})} ⊗ {∑ t^{^{q−m2} d} e^{q}(x^{2}, P^{2})} to complete the proof of (a) and (b). We note that the multiplicative prop- erty (b) is a direct consequence of the identity e^{−t(a+b)} = e^{−ta}e^{−tb}; it was for this reason we worked with the heat equation. Had we worked instead with the zeta function to study Tr(P^{−s}) the corresponding multiplicative property would have been much more difficult to derive.

We prove (c) as follows: expand p(x, ξ) = ∑_{j} p_{j}(x, ξ) into homogeneous polynomials where p_{j}(x, ξ) = ∑_{|α|=j} p_{α}(x)ξ^{α}. Suppose for the moment that p_{d}(x, ξ) = p_{d}(x, ξ)I_{V} is scalar so it commutes with every matrix. The approximate resolvant is given by: r0 = (p^{d}(x, ξ) − λ)−1 rn = −r_{0} _{∑ |α|+j+d−k=n j<n} d_{α ξ}r_{j}D_{α x}p_{k}/α! . Since p_{d} is scalar, r_{0} is scalar so it and all its derivatives commute with any matrix. If we assume inductively r_{j} is of order j in the jets of the symbol, then d_{α ξ}r_{j}D_{α x}p_{k}/α! is homogeneous of order j + |α| + d − k = n in the jets of the symbol. We observe inductively we can decompose r_{n} in the form: r^{n} = ^{∑ n=dj+d−|α| |α|≤n} r^{j 0}r^{n,j,α}(x)ξα where the r_{n,j,α}(x) are certain non-commutative polynomials in the jets of the total symbol of P which are homogeneous of order n in the sense we have defined. The next step in the proof of Lemma 1.7.4 was to define: e_{n}(t, x, ξ) = 1 2πi ∫_{γ} e^{−tλ}r_{n}(x, ξ, λ) dλ = 1 2πi _{∑ j,α} r_{n,j,α}(x)ξ^{α} ∫_{γ} e^{−tλ}r_{j 0} dλ e_{n}(x) = 1 2πi _{∑ j,α} r_{n,j,α}(x) ∫ (∫_{γ} e^{−λ}r_{j 0} dλ)ξ^{α} dξ. We note again that e_{n} = 0 if n is odd since the resulting function of ξ would be odd. The remaining coefficients of r_{n,j,α}(x) depend smoothly on the leading symbol p_{d}. This completes the proof of the lemma. If the leading symbol is not scalar, then the situation is more com- plicated. Choose a local frame to represent the p_{α} = p_{abα} as matri- ces. Let h(x, ξ, λ) = det(p_{d}(x, ξ) − λ)^{−1}. By Cramer’s rule, we can ex- press r_{0} = r_{0ab} = {(p_{d} − λ)^{−1}}_{ab} as polynomials in the {h, λ, ξ, p_{abα}(x)} variables. We noted previously that r_{n} was a sum of terms of the form r_{0}q_{0}r_{0}q_{1} … r_{0}q_{k}r_{0}. The matrix components of such a product can in turn be decomposed as a sum of terms h^{v}q̃v where q̃v is a polynomial in the (λ, ξ, p_{abα/β}) variables. The same induction argument which was used in the scalar case shows the q̃v will be homogeneous of order n in the jets of 58 1.7. Local Formula for the Index the symbol. The remainder of the argument is the same; performing the dλ dξ integral yields a smooth function of the leading symbol as a coefficient of such a term, but this function is not in general rational of course. The scalar case is much simpler as we don’t need to introduce the components of the matrices explicitly (although the frame dependence is still there of course) since r_{0} can be commuted. This is a technical point, but one often useful in making specific calculations. We define the scalar invariants a_{n}(x, P) = Tr e_{n}(x, P) where the trace is the fiber trace in V over the point x. These scalar invariants a_{n}(x, P) inherit suitable functorial properties from the functorial properties of the invariants e_{n}(x, P). It is immediate that: Tr_{L2} e−tP = ∫_{M} Tr_{Vx} K(t, x, x) dvol(x) ∼ _{∞ ∑ n=0} t^{^{m−n} d} ∫_{M} a_{n}(x, P) dvol(x). ∼ _{∞ ∑ n=0} t^{^{m−n} d} a_{n}(P) where a_{n}(P) = ∫_{M} a_{n}(x, P) dvol(x) is the integrated invariant. This is a spectral invariant of P which can be computed from local information about the symbol of P. Let (P, V) be an elliptic complex of differential operators and let Δ_{i} be the associated Laplacians. We define: a_{n}(x, P) = _{∑ i}(−1)^{i} Tr e_{n}(x, Δ_{i}) then Lemma 1.6.5 and the remark which follows this lemma imply index(P) = _{∑ i}(−1)^{i} Tr e^{−tΔi} ∼ _{∞ ∑ n=0} t^{^{n−m} d} ∫_{M} a_{n}(x, P) dvol(x). Since the left hand side does not depend on the parameter t, we conclude: Theorem 1.7.6. Let (P, V) be a elliptic complex of differential opera- tors. (a) a_{n}(x, P) can be computed in any coordinate system and relative to any local frames as a complicated combinatorial expression in the jets of P and of P^{∗} up to some finite order. a_{n} = 0 if n is odd.

(b)  ∫^{M} a^{n}(x, P) dvol(x) = \{ index(P) if n = m 0  if n ≠ m . We note as an immediate consequence that index(P) = 0 if m is odd. The local nature of the invariants a_{n} will play a very important role in our discussion of the index theorem. We will develop some of their functorial properties at that point. We give one simple example to illustrate this fact. Let π: M_{1} → M_{2} be a finite covering projection with fiber F. We can choose the metric on M_{1} to be the pull-back of a metric on M_{2} so that π is an isometry. Let d be the operator of the de Rham complex. a_{n}(x, d) = a_{n}(πx, d) since a_{n} is locally defined. This implies: χ(M_{1}) = ∫_{M1} a_{m}(x, d) dvol(x) = |F | ∫_{M2} a_{m}(x, d) dvol(x) = |F |χ(M_{2}) so the Euler-Poincare characteristic is multiplicative under finite coverings. A similar argument shows the signature is multiplicative under orientation preserving coverings and that the arithmetic genus is multiplicative under holomorphic coverings. (We will discuss the signature and arithmetic genus in more detail in Chapter 3). We conclude this section with a minor generalization of Lemma 1.7.5 which will be useful in discussing the eta invariant in section 1.10: Lemma 1.7.7. Let P: C^{∞}(V) → C^{∞}(V) be an elliptic self-adjoint partial differential operator of order d > 0 with positive definite leading symbol and let Q be an auxilary partial differential operator on C^{∞}(V) of order a ≥ 0. Qe^{−tP} is an infinitely smoothing operator with kernel QK(t, x, y). There is an asymptotic expansion on the diagonal: \{QK(t, x, y)\}|_{x=y} ∼ _{∞ ∑ n=0} t^{(n−m−a)/d} e_{n}(x, Q, P). The e_{n} are smooth local invariants of the jets of the symbols of P and Q and e_{n} = 0 if n + a is odd. If we let Q = ∑ q_{α}D_{α x}, we define ord(q_{α}) = a − |α|. If the leading symbol of P is scalar, we can compute e_{n}(x, Q, P) as a non-commutative polynomial in the variables \{q_{α}, p_{α/β}\} which is homo- geneous of order n in the jets with coefficients which depend smoothly on the \{p_{α}\}_{|α|=d} variables. This expression is \underline{linear} in the \{q_{α}\} variables and does not involve the higher jets of these variables. If the leading symbol of P is not scalar, there is a similar expression for the matrix components of e_{n} in the matrix components of these variables. e_{n}(x, Q, P) is additive and multiplicative in the sense of Lemma 1.7.5(a) and (b) with respect to direct sums of operators and tensor products over product manifolds. Remark: In fact it is not necessary to assume P is self-adjoint to define e−tP and to define the asymptotic series. It is easy to generalize the techniques

60 1.7. Local Formula for the Index we have developed to prove this lemma continues to hold true if we only asume that det(p_{d}(x, ξ) − λ) ≠ 0 for ξ ≠ 0 and Im(λ) ≤ 0. This implies that the spectrum of P is pure point and contained in a cone about the positive real axis. We omit the details. Proof: Qe^{−tP} has smooth kernel QK(t, x, y) where Q acts as a differen- tial operator on the x variables. One representative piece is: q_{α}(x)D_{α x}K_{n}(t, x, y) = q_{α}(x)D_{α x} ∫ e^{i(x−y)·ξ} e_{n}(t, x, ξ) dξ = q_{α}(x)_{∑ β+γ=α} α! β! γ! ∫ e^{i(x−y)·ξ} ξ^{β}D_{γ x}e_{n}(t, x, ξ) dξ. We define q_{β,γ}(x) = q_{α}(x) α! β! γ! where β + γ = α, then a representative term of this kernel has the form: q_{β,γ}(x) ∫ e^{i(x−y)·ξ} ξ^{β}D_{γ x}e_{n}(t, x, ξ) dξ where q_{β,γ}(x) is homogeneous of order a−|β|−|γ| in the jets of the symbol of Q. We suppose the leading symbol is scalar to simplify the computations; the general case is handled similarly using Cramer’s rule. We express e_{n} = 1 2πi ∫ e^{−tλ}r_{n}(x, ξ, λ) dλ where r_{n} is a sum of terms of the form r_{n,j,α}(x)ξ^{α}r_{j 0}. When we differentiate such a term with respect to the x variables, we change the order in the jets of the symbol, but do not change the (ξ, λ) degree of homogeneity. Therefore QK(t, x, y) is a sum of terms of the form: q_{β,γ}(x) ∫ {∫ e^{−tλ}r_{j 0}(x, ξ, λ) dλ} e^{i(x−y)·ξ} ξ^{β}ξ^{δ}r̃n,j,δ(x) dξ where: r_{n,j,δ} is of order n + |γ| in the jets of the symbol, − d − n = −jd + |δ|. We evaluate on the diagonal to set x = y. This expression is homogeneous of order −d − n + |β| in (ξ, λ), so when we make the appropriate change of variables we calculate this term becomes: t^{(n−m−|β|)/d} q^{β,γ}(x)^{r̃n,j,δ}(x) · {∫ {e^{−λ}r^{j 0}(x, ξ, λ) dλ} ξ^{β}ξ^{δ} dξ} .

Of an Elliptic Operator 61

This is of order n+|γ|+a−|β|−|γ| = n+a−|β| = ν in the jets of the symbol. The exponent of t is (n−m−|β|)/d = (ν−a+|β|−m−|β|)/d = (ν−a−m)/d. The asymptotic formula has the proper form if we index by the order in the jets of the symbol ν. It is linear in q and vanishes if |β| + |δ| is odd. As d is even, we compute: ν + a = n + a − |β| + a = jd − d − |δ| + 2a − |β| ≡ |δ| + |β| mod 2 so that this term vanishes if ν + a is odd. This completes the proof.

1.8. Lefschetz Fixed Point Theorems.

Let T: M → M be a continuous map. T^{∗} defines a map on the cohomology of M and the Lefschetz number of T is defined by:

L(T) = ∑(−1)^{p} Tr(T^{∗} on H^{p}(M; C)).

This is always an integer. We present the following example to illustrate the concepts involved. Let M = S^{1} × S^{1} be the two dimensional torus which we realize as R^{2} modulo the integer lattice Z^{2}. We define T (x, y) = (n_{1}x + n_{2}y, n_{3}x + n_{4}y) where the n_{i} are integers. Since this preserves the integer lattice, this defines a map from M to itself. We compute:

T^{∗}(1) = 1,  T^{∗}(dx ∧ dy) = (n_{1}n_{4} − n_{2}n_{3})dx ∧ dy T^{∗}(dx) = n_{1} dx + n_{2} dy,  T^{∗}(dy) = n_{3} dx + n_{4} dy L(T) = 1 + (n_{1}n_{4} − n_{2}n_{3}) − (n_{1} + n_{4}).

Of course, there are many other interesting examples.

We computed L(T) in the above example using the de Rham isomorphism. We let T^{∗} = Λ^{p}(dT): Λ^{p}(T^{∗}M) → Λ^{p}(T^{∗}M) to be the pull-back operation. It is a map from the fiber over T (x) to the fiber over x. Since dT^{∗} = T^{∗}d, T^{∗} induces a map on H^{p}(M, C) = ker(d_{p})/ image(d_{p−1}). If T is the identity map, then L(T) = χ(M). The perhaps somewhat surprising fact is that Lemma 1.6.5 can be generalized to compute L(T) in terms of the heat equation.

Let Δ_{p} = (dδ + δd)_{p}: C^{∞}(Λ^{p}T^{∗}M) → C^{∞}(Λ^{p}T^{∗}M) be the associated Laplacian. We decompose L^{2}(Λ^{p}T^{∗}M) = ⨁_{λ} E_{p}(λ) into the eigenspaces of Δ_{p}. We let π(p, λ) denote orthogonal projection on these subspaces, and we define T^{∗}(p, λ) = π(p, λ)T^{∗}: E_{p}(λ) → E_{p}(λ). It is immediate that:

Tr(T^{∗}e^{−tΔp}) = _{∑ λ} e^{−tλ} Tr(T^{∗}(p, λ)).

Since dT^{∗} = T^{∗}d and dπ = πd, for λ ≠ 0 we get a chain map between the exact sequences:

⋯ E_{p−1}(λ) ^{d} ⟶ E_{p}(λ) ^{d} ⟶ E_{p+1}(λ) ⋯
T^∗  T^∗  T^∗
⋯ L^{2}(Λ^{p−1}) ^{d} ⟶ L^{2}(Λ^{p}) ^{d} ⟶ L^{2}(Λ^{p+1}) ⋯
π  π  π
⋯ E_{p−1}(λ) ^{d} ⟶ E_{p}(λ) ^{d} ⟶ E_{p+1}(λ) ⋯

Lefschetz Fixed Point Theorems 63

Since this diagram commutes and since the two rows are long exact se- quences of finite dimensional vector spaces, a standard result in homological algebra implies: _{∑ p}(−1)^{p} Tr(T^{∗}(p, λ)) = 0 for λ ≠ 0. It is easy to see the corresponding sum for λ = 0 yields L(T), so Lemma 1.6.5 generalizes to give a heat equation formula for L(T). This computation was purely formal and did not depend on the fact that we were dealing with the de Rham complex. Lemma 1.8.1. Let (P, V) be an elliptic complex over M and let T: M → M be smooth. We assume given linear maps V_{i}(T): V_{i}(T x) → V_{i}(x) so that P_{i}(V_{i}(T)) = V_{i}(T)P_{i}. Then T induces a map on H^{p}(P, V). We define L(T)_{P} = _{∑ p}(−1)^{p} Tr(T on H^{p}(P, V)). Then we can compute L(T)_{P} = ∑_{p} Tr(V_{i}(T)e^{−tΔp}) where Δ_{p} = (P^{∗}P + P P^{∗})_{p} is the associated Laplacian. The V_{i}(T) are a smooth linear action of T on the bundles V_{i}. They will be given by the representations involved for the de Rham, signature, spin, and Dolbeault complexes as we shall discuss in Chapter 4. We usually denote V_{i}(T) by T^{∗} unless it is necessary to specify the action involved. This Lemma implies Lemma 1.6.5 if we take T = I and V_{i}(T) = I. To generalize Lemma 1.7.4 and thereby get a local formula for L(T)_{P}, we must place some restrictions on the map T. We assume the fixed point set of T consists of the finite disjoint union of smooth submanifolds N_{i}. Let ν(N_{i}) = T (M)/T (N_{i}) be the normal bundle over the submanifold N_{i}. Since dT preserves T (N_{i}), it induces a map dT_{ν} on the bundle ν(N_{i}). We suppose det(I − dT_{ν}) ≠ 0 as a non-degeneracy condition; there are no normal directions left fixed infinitesimally by T. If T is an isometry, this condition is automatic. We can construct a non-example by defining T (z) = z/(z + 1): S^{2} → S^{2}. The only fixed point is at z = 0 and dT (0) = I, so this fixed point is degenerate. If K is the kernel of e^{−tP}, we pull back the kernel to define T^{∗}(K)(t, x, y) = T^{∗}(x)K(t, T x, y). It is immediate T^{∗}K is the kernel of T^{∗}e^{−tP}. Lemma 1.8.2. Let P be an elliptic partial differential operator of order d > 0 which is self-adjoint and which has a positive definite leading symbol for ξ ≠ 0. Let T: M → M be a smooth non-degenerate map and let T^{∗}: V_{T (x)} → V_{x} be a smooth linear action. If K is the kernel of e^{−tP} then Tr(T^{∗}e^{−tP}) = ∫_{M} Tr(T^{∗}K)(t, x, x) dvol(x). Furthermore: (a) If T has no fixed points, | Tr(T^{∗}e^{−tP})| ≤ C_{n}t^{−n} as t → 0^{+} for any n.

64 1.8. Lefschetz Fixed Point Theorems

(b) If the fixed point set of T consists of submanifolds N_{i} of dimension m_{i}, we will construct scalar invariants a_{n}(x) which depend functorially upon a finite number of jets of the symbol and of T. The a_{n}(x) are defined over N_{i} and Tr(T^{∗}e^{−tP}) ∼ _{∑ i}_{∞ ∑ n=0} t^{^{n−mi} d} ∫_{Ni} a_{n}(x) dvol_{i}(x). dvol_{i}(x) denotes the Riemannian measure on the submanifold. It follows that if T has no fixed points, then L(T)_{P} = 0. Proof: Let {e_{n}, r_{n}, K_{n}} be as defined in section 1.7. The estimates of Lemma 1.7.3 show that |T^{∗}K − ∑^{n≤n0} T^{∗}K^{n}|^{∞,k} ≤ C(k)t^{−k} as t → 0+ for any k if n_{0} = n_{0}(k). We may therefore replace K by K_{n} in proving (a) and (b). We recall that: e_{n}(t, x, ξ) = 1 2πi ∫_{γ} e^{−tλ}r_{n}(x, ξ, λ) dλ. We use the homogeneity of r_{n} to express: e_{n}(t, x, ξ) = 1 2πi ∫_{γ} e^{−λ}r_{n}(x, ξ, t^{−1}λ)t^{−1} dλ = t^{^{n} d} 1 2πi ∫_{γ} e^{−λ}r_{n}(x, t^{^{1} d} ξ, λ) dλ = t^{^{n} d} e_{n}(x, t^{^{1} d} ξ) where we define e_{n}(x, ξ) = e_{n}(1, x, ξ) ∈ S^{−∞}. Then: K_{n}(t, x, y) = ∫ e^{i(x−y)·ξ} e_{n}(t, x, ξ) dξ = t^{^{n−m} d} ∫ e^{i(x−y)·t^{−^{1} d} ξ} e_{n}(x, ξ) dξ. This shows that: T^{∗}K_{n}(t, x, x) = t^{^{n−m} d} ∫ T^{∗}(x)e^{i(T x−x)·t^{−1/d}ξ} e_{n}(T x, ξ) dξ. We must study terms which have the form: ∫ e^{i(T x−x)·t^{−1/d}ξ} Tr(T^{∗}(x)e_{n}(T x, ξ)) dξ dx where the integral is over the cotangent space T^{∗}(M). We use the method of stationary phase on this highly oscillatory integral. We first bound

|T x − x| ≥ ε > 0. Using the argument developed in Lemma 1.2.6, we integrate by parts to bound this integral by C(n, k, ε)t^{k} as t → 0 for any k. If T has no fixed points, this proves (a). There is a slight amount of notational sloppiness here since we really should introduce partitions of unity and coordinate charts to define T x − x, but we supress these details in the interests of clarity. We can localize the integral to an arbitrarily small neighborhood of the fixed point set in proving (b). We shall assume for notational simplicity that the fixed point set of T consists of a single submanifold N of dimension m_{1}. The map dT_{ν} on the normal bundle has no eigenvalue 1. We identify ν with the span of the generalized eigenvectors of dT on T (M)|_{N} which correspond to eigenvalues not equal to 1. This gives a direct sum decomposition over N:  T (M) = T (N) ⊕ ν  and dT = I ⊕ dT_{ν}. We choose a Riemannian metric for M so this splitting is orthogonal. We emphasize that these are bundles over N and not over the whole manifold M.We describe the geometry near the fixed manifold N using the nor- mal bundle ν. Let y = (y_{1},…, y_{m1} ) be local coordinates on N and let \{s1,…, sm−m1 \} be a local orthonormal frame for ν. We use this local orthonormal frame to introduce fiber coordinates z = (z_{1},…, z_{m−m1} ) for ν by decomposing any s ∈ ν in the form: s = _{∑ j} z_{j}sj(y). We let x = (y, z) be local coordinates for ν. The geodesic flow identifies a neighborhood of the zero section of the bundle ν with a neighborhood of N in M so we can also regard x = (y, z) as local coordinates on M. We decompose  T (x) = (T_{1}(x), T_{2}(x)) into tangential and fiber coordinates. Because the Jacobian matrix has the form:  dT (y, 0) = ( {I 0 \atop 0 dTν} ) we conclude that T_{1}(x) − y must vanish to second order in z along N. We integrate Tr(T^{∗}K_{n})(t, x, x) along a small neighborhood of the zero section of ν. We shall integrate along the fibers first to reduce this to an integral along N. We decompose ξ = (ξ_{1}, ξ_{2}) corresponding to the decomposition of x = (y, z). Let D(ν) = \{ (y, z) : |z| ≤ 1 \}

66 1.8. Lefschetz Fixed Point Theorems be the unit disk bundle of the normal bundle. Let U = T^{∗}(D(ν)) be the cotangent bundle of the unit disk bundle of the normal bundle. We assume the metric chosen so the geodesic flow embeds D(ν) in M. We parametrize U by { (y, z, ξ_{1}, ξ_{2}) : |z| ≤ 1 }. Let s = t^{^{1} d}. Modulo terms which vanish to infinite order in s, we compute:^{I ^{def} = ∫_{M} Tr((T^{∗}K_{n})(t, x, x)) dx = sn−m ∫U ei(T^{1}(y,z)−y)·ξ^{1}s−1 ei(T^{2}(y,z)−z)·ξ^{2}s−1} × Tr(T^{∗}(x)e_{n}(T x, ξ_{1}, ξ_{2})) dξ_{1} dξ_{2} dz dy. The non-degeneracy assumption on T means the phase function w = T_{2}(y, z)−z defines a non-degenerate change of variables if we replace (y, z) by (y, w). This transforms the integral into the form: I = s^{n−m} ∫^{U} e^{i(T_{1}(y,w)−y)·ξ_{1}s−1} ei^{w}·ξ^{2}s−1 × | det(I − dT_{2})|^{−1} Tr(T^{∗}(y, w)e_{n}(T (y, w), ξ_{1}, ξ_{2}) dξ_{1} dξ_{2} dw dy, where U is the image of U under this change. We now make another change of coordinates to let w = s^{−1}w. As s → 0, s^{−1}U will converge to T^{∗}(v). Since dw = s^{m−m1} dw this transforms the integral I to the form: I = s^{n−m1} ∫_{s^{−1}U} e^{i(T_{1}(y,sw)−y)·ξ_{1}s−1} e^{iw·ξ2} | det(I − dT_{2})^{−1}|(y, sw) × Tr(T^{∗}(y, sw)e_{n}(T (y, sw), ξ_{1}, ξ_{2})) dξ_{1} dξ_{2} dw dy. Since the phase function T_{1}(y, w) − y vanishes to second order at w = 0, the function (T_{1}(y, sw) − y)s^{−1} is regular at s = 0. Define: e_{′n}(y, w, ξ_{1}, ξ_{2}, s) = e^{i(T_{1}(y,sw)−y)·ξ_{1}s−1} | det(I − dT_{2})(y, sw)| × Tr(T^{∗}(y, sw)e_{n}(T (y, sw), ξ_{1}, ξ_{2})). This vanishes to infinite order in ξ_{1}, ξ_{2} at ∞ and is regular at s = 0. To complete the proof of Lemma 1.8.2, we must evaluate: s^{(n−m_{1})} ∫_{s^{−1}U} e_{′n}(y, w, ξ_{1}, ξ_{2}, s)e^{iw·ξ2} dξ_{1} dξ_{2} dw dy. We expand e_{′n} in a Taylor series in s centered at s = 0. If we differentiate e_{′n} with respect to s a total of k times and evaluate at s = 0, then the exponential term disappears and we are left with an expression which is

Lefschez Fixed Point Theorems 67

polynomial in w and of degree at worst 2k. It still vanishes to infinite order in (ξ_{1}, ξ_{2}). We decompose: e_{′n} = _{∑ j≤j0} s^{j} _{∑ |α|≤2j} c_{j,α}(y, ξ_{1}, ξ_{2})w^{α} + s^{j0}ε(y, w, ξ_{1}, ξ_{2}, s) where ε is the remainder term. We first study a term of the form: s^{n+j−m1} ∫_{|w|<s−1} c_{j,α}(y, ξ_{1}, ξ_{2})w^{α}e^{iξ_{2}·w} dξ_{1} dξ_{2} dw dy. Since c_{j,α} vanishes to infinite order in ξ, the dξ integral creates a function which vanishes to infinite order in w. We can let s → 0 to replace the domain of integration by the normal fiber. The error vanishes to infinite order in s and gives a smooth function of y. The dw integral just yields the inverse Fourier transform with appropriate terms and gives rise to asymp- totics of the proper form. The error term in the Taylor series grows at worst polynomially in w and can be bounded similarly. This completes the proof. If there is more than one component of the fixed point set, we sum over the components since each component makes a separate contibution to the asymptotic series. The case of isolated fixed points is of particular interest. Let d = 2 and σ_{L}(P) = |ξ|^{2}I_{V}. We compute the first term at a fixed point: r^{0}(x, ξ, λ) = (|ξ|^{2} − λ)^{−1} e^{0}(x, ξ, t) = e−t|ξ|2 ∫ Tr(T^{∗}K_{0})(t, x, x) dvol(x) = ∫ Tr(T^{∗}(x))e^{i(T x−x)·ξ} e^{−t|ξ|2} dξ dx. We assume T (0) = 0 is an isolated non-degenerate fixed point. We let y = T x − x be a change of variables and compute the first term: ∫ Tr(T^{∗}(y))e^{iy·ξ} | det(I − dT (y))|^{−1}e^{−t|ξ|2} dξ dy. We make a change of variables ξ → ξt^{−1/2} and y → yt^{1/2} to express the first term: ∫ Tr(T^{∗}(yt^{1/2}))e^{iy·ξ} | det(I − dT (yt^{1/2})|e^{−|ξ|2} dξ dy. The dξ integral just gives e^{−|y|2} so this becomes ∫ Tr(T^{∗}(yt^{1/2}))| det(I − dT (yt^{1/2}))|^{−1}e^{−|y|2} dy. We expand this in a Taylor series at t = 0 and evaluate to get Tr T^{∗}(0) | det(I − dT (O))|^{−1}. This proves:

68 1.8. Lefschetz Fixed Point Theorems Lemma 1.8.3. Let P be a second order elliptic partial differential op- erator with leading symbol |ξ|^{2}I. Let T: M → M be smooth with non- degenerate isolated fixed points. Then: Tr T^{∗}e^{−tP} = _{∑ i} Tr(T^{∗}) | det(I − dT)|^{−1}(x_{i}) summed over the fixed point set. We combine Lemmas 1.8.1 and 1.8.2 to constuct a local formula for L(T)_{P} to generalize the local formula for index(P) given by Theorem 1.7.6; we will discuss this further in the fourth chapter. We can use Lemma 1.8.3 to prove the classical Lefschetz fixed point formula for the de Rham complex. Let T: V → V be a linear map, then it is easily computed that:_{∑ p}(−1)^{p} Tr Λ^{p}(T) = det(I − T). We compute: L(T) = _{∑ p}(−1)^{p} Tr(T^{∗}e^{−tΔp}) = _{∑ p,i}(−1)^{p} Tr(Λ^{p}dT) | det(I − dT)|^{−1}(x_{i}) = _{∑ i} det(I − dT) | det(I − dT)|^{−1}(x_{i}) = _{∑ i} sign det(I − dT)(x_{i}) summed over the fixed point set of T. This proves: Theorem 1.8.4 (Classical Lefschetz Fixed Point Formula). Let T: M → M be smooth with isolated non-degenerate fixed points. Then: L(T) = _{∑ p}(−1)^{p} Tr(T^{∗} on H^{p}(M; C)) = _{∑ i} sign det(I − dT)(x_{i}) summed over the fixed point set. Remark: We can generalize Lemma 1.8.2 to study Tr(T^{∗}Qe^{−tP}) where Q is an auxilary differential operator of order a. Just as in lemma 1.7.7 we may obtain an asymptotic series: Tr(T^{∗}Qe^{−tP}) ∼ _{∑ i}_{∞ ∑ n=0} t^{(n−m_{i}−a)/d} a^{n}(x, Q, P) dvol^{i}(x).

Lefschez Fixed Point Theorems 69

We shall omit the details as the additional terms created by the operator Q are exactly the same as those given in the proof of Lemma 1.7.7. Each term a_{n} is homogeneous of order n in the jets of the symbols of (Q, P) and of the map T in a suitable sense.

1.9. Elliptic Boundary Value Problems. In section 1.7 we derived a local formula for the index of an elliptic partial-differential complex using heat equation methods. This formula will lead to a heat equation proof of the Atiyah-Singer index theorem which we shall discuss later. Unfortunately, it is not known at present how to give a heat equation proof of the Atiyah-Bott index theorem for manifolds with boundary in full generality. We must adopt a much stronger notion of ellipticity to deal with the analytic problems involved. This will yield a heat equation proof of the Gauss-Bonnet theorem for manifolds with boundary which we shall discuss in the fourth chapter. Let M be a smooth compact manifold with smooth boundary dM and let P: C^{∞}(V) → C^{∞}(V) be a partial differential operator of order d > 0. We let p = σ_{L}(P) be the leading symbol of P. We assume henceforth that p is self-adjoint and elliptic—i.e., det p(x, ξ) ≠ 0 for ξ ≠ 0. Let R_{±} denote the non-zero positive/negative real numbers. It is immediate that det{p(x, ξ) − λ} ≠ 0  for (ξ, λ) ≠ (0, 0) ∈ T^{∗}(M) × {C − R_{+} − R_{−}}, since p is self-adjoint and elliptic. We fix a fiber metric on V and a volume element on M to define the global inner product (·, ·) on L^{2}(V). We assume that P is formally self- adjoint:  (P f, g) = (f, P g) for f and g smooth section with supports disjoint from the boundary dM. We must impose boundary conditions to make P self-adjoint. For example, if P = −∂^{2}/∂x^{2} on the line segment [0, A], then P is formally self-adjoint and elliptic, but we must impose Neumann or Dirichlet boundary condi- tions to ensure that P is self-adjoint with discrete spectrum. Near dM we let x = (y, r) where y = (y_{1},…, y_{m−1}) is a system of local coordinates on dM and where r is the normal distance to the boundary. We assume dM = {x : r(x) = 0} and that ∂/∂r is the inward unit normal. We further normalize the choice of coordinate by requiring the curves x(r) = (y_{0}, r) for r ∈ [0, δ) are unit speed geodesics for any y_{0} ∈ dM. The inward geodesic flow identifies a neighborhood of dM in M with the collar dM × [0, δ) for some δ > 0. The collaring gives a splitting of T (M) = T (dM) ⊕ T (R) and a dual splitting T^{∗}(M) = T^{∗}(dM) ⊕ T^{∗}(R). We let ξ = (ζ, z) for ζ ∈ T^{∗}(dM) and z ∈ T^{∗}(R) reflect this splitting. It is convenient to discuss boundary conditions in the context of graded vector bundles. A graded bundle U over M is a vector bundle U together with a fixed decompositionU = U_{0} ⊕ ⋯ ⊕ Ud−1 into sub-bundles U_{j} where U_{j} = {0} is permitted in this decomposition. We let W = V ⊗ 1^{d} = V ⊕ ⋯ ⊕ V restricted to dM be the bundle of

Cauchy data. If W_{j} = V |_{dM} is the (j + 1)^{st} factor in this decomposition, then this defines a natural grading on W; we identify W_{j} with the bundle of j^{th} normal derivatives. The restriction map: \underline{γ}: C^{∞}(V) → C^{∞}(W) defined by: \underline{γ}(f) = (f^{0},…, f^{d−1})  where f^{j} = D^{j r}f|^{dM} = (−i)^{j} ∂^{j}f ∂rj ||||dM assigns to any smooth section its Cauchy data. Let W′ be an auxiliary graded vector bundle over dM. We assume that dim W = d · dim V is even and that 2 dim W′ = dim W. Let B: C^{∞}(W) → C^{∞}(W′) be a tangential differential operator over dM. Decompose B = B_{ij} for  B_{ij}: C^{∞}(W_{i}) → C^{∞}(W_{′ j}) and assume that  ord(B_{ij}) ≤ j − i. It is natural to regard a section to C^{∞}(W_{i}) as being of order i and to define the graded leading symbol of B by: σ^{g}(B)^{ij}(y, ζ) = \{ σ_{L}(B_{ij})(y, ζ) if ord(B_{ij}) = j − i 0  if ord(B^{ij}) < j − i. We then regard σ_{g}(B) as being of graded homogeneity 0. We let P_{B} be the operator P restricted to those f ∈ C^{∞}(V) such that B\underline{γ}f = 0. For example, let P = −∂^{2}/∂x^{2} on the interval [0, A]. To define Dirichlet boundary conditions at x = 0, we would set: W′ = C ⊕ 0  and B_{0,0} = 1, B_{1,1} = B_{0,1} = B_{1,0} = 0 while to define Neumann boundary conditions at x = 0, we would set: W′ = 0 ⊕ C  and B_{1,1} = 1, B_{0,0} = B_{0,1} = B_{1,0} = 0. To define the notion of ellipticity we shall need, we consider the ordinary differential equation: p(y, 0, ζ, D_{r})f(r) = λf(r)  with _{lim r→∞} f(r) = 0 where  (ζ, λ) ≠ (0, 0) ∈ T^{∗}(dM) × C − R_{+}.

72 1.9. Elliptic Boundary We say that (P, B) is elliptic with respect to C−R_{+} if det(p(x, ξ)−λ) ≠ 0 on the interior for all (ζ, λ) ≠ (0, 0) ∈ T^{∗}(M) × C − R_{+} and if on the boundary there always exists a unique solution to this ordinary differential equation such that σ_{g}(B)(y, ζ)\underline{γ}f = f′ for any prescribed f′ ∈ W′. In a similar fashion, we define ellipticity with respect to C − R_{+} − R_{−} if these conditions hold for λ ∈ C − R_{+} − R_{−}. Again, we illustrate these notions for the operator P = −∂^{2}/∂x^{2} and a boundary condition at x = 0. Since dim M = 1, there is no dependence on ζ and we must simply study the ordinary differential equation: −f′′ = λf with _{lim r→∞} f(r) = 0 (λ ≠ 0). If λ ∈ C − R_{+}, then we can express λ = μ^{2} for Im(μ) > 0. Solutions to the equation −f′′ = λf are of the form f(r) = ae^{iμr} + be^{−iμr}. The decay at ∞ implies b = 0 so f(r) = ae^{iμr}. Such a function is uniquely determined by either its Dirichlet or Neumann data at r = 0 and hence P is elliptic with respect to either Neumann or Dirichlet boundary conditions. We assume henceforth that P_{B} is self-adjoint and that (P, B) is elliptic with respect to either the cone C − R_{+} or the cone C − R_{+} − R_{−}. It is beyond the scope of this book to develop the analysis required to discuss elliptic boundary value problems; we shall simply quote the required results and refer to the appropriate papers of Seeley and Greiner for further details. Lemmas 1.6.3 and 1.6.4 generalize to yield: Lemma 1.9.1. Let P: C^{∞}(V) → C^{∞}(V) be an elliptic partial differential operator of order d > 0. Let B be a boundary condition. We assume (P, B) is self-adjoint and elliptic with respect to C − R_{+} − R_{−}. (a) We can find a complete orthonormal system \{ϕ_{n}\}_{∞ n=1} for L^{2}(V) with P ϕ_{n} = λ_{n}ϕ_{n}. (b) ϕ_{n} ∈ C^{∞}(V) and satisfy the boundary condition B\underline{γ}ϕ_{n} = 0. (c) λ_{n} ∈ R and lim_{n→∞} |λ_{n}| = ∞. If we order the λ_{n} so |λ_{1}| ≤ |λ_{2}| ⋯ then there exists n_{0} and δ > 0 so that |λ_{n}| > n^{δ} for n > n_{0}. (d) If (P, B) is elliptic with respect to the cone C − R_{+}, then the λ_{n} are bounded from below and spec(P_{B}) is contained in [−C, ∞) for some C. For example, if P = − ∂2 ∂x2 on [0, π] with Dirichlet boundary condi- tions, then the spectral resolution becomes \{_{√ 2 π} sin(nx)\}_{∞ n=1} and the cor- responding eigenvalues are n^{2}. If (P, B) is elliptic with respect to the cone C − R_{+}, then e^{−tPB} is a smoothing operator with smooth kernel K(t, x, x′) defined by: K(t, x, x′) = _{∑ n} e^{−tλn} ϕ_{n}(x) ⊗ ϕ̄n(x′). Lemma 1.7.4 generalizes to yield:

Value Problems 73

Lemma 1.9.2. Let (P, B) be elliptic with respect to the cone C − R_{+}, and be of order d > 0. (a) e^{−tPB} is an infinitely smoothing operator with smooth kernel K(t, x, x′). (b) On the interior we define a_{n}(x, P) = Tr e_{n}(x, P) be as in Lemma 1.7.4. On the boundary we define a_{n}(y, P, B) using a complicated combinatorial recipe which depends functorially on a finite number of jets of the symbols of P and of B. (c) As t → 0^{+} we have an asymptotic expansion: Tr e^{−tPB} = _{∑ n} e−tλn = ∫_{M} Tr K(t, x, x) dvol(x) ∼ _{∞ ∑ n=0} t^{^{n−m} d} ∫_{M} a_{n}(x, P) dvol(x) +_{∞ ∑ n=0} t^{^{n−m+1} d} ∫_{dM} a_{n}(y, P, B) dvol(y). (d) a_{n}(x, P) and a_{n}(y, P, B) are invariantly defined scalar valued functions which do not depend on the coordinate system chosen nor on the local frame chosen. a_{n}(x, P) = 0 if n is odd, but a_{n}(y, P, B) is in general non-zero for all values of n. The interior term a_{n}(x, P) arises from the calculus described previously. We first construct a parametrix on the interior. Since this parametrix will not have the proper boundary values, it is necessary to add a boundary correction term which gives rise to the additional boundary integrands a_{n}(y, P, B). To illustrate this asymptotic series, we let P = − ∂2 ∂x2 − e(x) on the in- terval [0, A] where e(x) is a real potential. Let B be the modified Neumann boundary conditions: f′(0) + s(0)f(0) = f′(A) + s(A)f(A) = 0. This is el- liptic and self-adjoint with respect to the cone C−R_{+}. It can be computed that:^{Tr e−tPB} ∼ (4πt)^{−1/2} ∫ {1 + t · e(x) + t^{2} · (e′′(x) + 3e(x)^{2})/6 + ⋯} dx + 1 2 + ( tπ )^{1/2} (s(0) − s(A)) + t4 · (e(0) + e(A) + 2s^{2}(0) + 2s^{2}(A)) + ⋯ The terms arising from the interior increase by integer powers of t while the terms from the boundary increase by powers of t^{1/2}. In this integral, dx is ordinary unnormalized Lebesgue measure.

In practice, we shall be interested in first order operators which are elliptic with respect to the cone C−R_{+} −R_{−} and which have both positive and negative spectrum. An interesting measure of the spectral asymmetry of such an operator is obtained by studying Tr(P_{B}e^{−tP^{2} B}) which will be discussed in more detail in the next section. To study the index of an elliptic operator, however, it suffices to study e^{−tP^{2} B}; it is necessary to work with P_{2 B} of course to ensure that the spectrum is positive so this converges to define a smoothing operator. There are two approaches which are available. The first is to work with the function e^{−tλ2} and integrate over a path of the form: [figure] to define e^{−tP^{2} B} directly using the functional calculus. The second approach is to define e^{−tP^{2} B} using the operator P^{2} with boundary conditions B\underline{γ}f = B\underline{γ}P f = 0. These two approaches both yield the same operator and give an appropriate asymptotic expansion which generalizes Lemma 1.9.2. We use the heat equation to construct a local formula for the index of certain elliptic complexes. Let Q: C^{∞}(V_{1}) → C^{∞}(V_{2}) be an elliptic differential operator of order d > 0. Let Q^{∗}: C^{∞}(V_{2}) → C^{∞}(V_{1}) be the formal adjoint. Let B_{1}: C^{∞}(W_{1}) → C^{∞}(W_{′ 1}) be a boundary condition for the operator Q. We assume there exists a boundary condition B^{2} for Q∗ of the same form so that (Q_{B1} )^{∗} = (Q^{∗})_{B2} . We form:  P = Q ⊕ Q^{∗}  and B = B_{1} ⊕ B2 so P: C^{∞}(V_{1} ⊕ V_{2}) → C^{∞}(V_{1} ⊕ V_{2}). Then P_{B} will be self-adjoint. We assume that (P, B) is elliptic with respect to the cone C − R_{+} − R_{−}. This is a very strong assmption which rules out many interesting cases, but is necessary to treat the index theorem using heat equation methods. We emphasize that the Atiyah-Bott theorem in its full generality does not follow from these methods.

Since P^{2} = Q^{∗}Q + QQ^{∗}, it is clear that P_{2 B} decomposes as the sum of two operators which preserve C^{∞}(V_{1}) and C^{∞}(V_{2}). We let S be the endomorphism +1 on V_{1} and −1 on V_{2} to take care of the signs; it is clear P_{2 B}S = S P_{2 B}. The same cancellation lemma we have used previously yields: index(Q_{1}, B_{1}) = dim N(Q_{B1} ) − dim N(Q_{∗ B2} ) = Tr(S e^{−tP^{2} B}). (The fact that this definition of the index agrees with the definition given in the Atiyah-Bott paper follows from the fact that B: C^{∞}(W) → C^{∞}(W_{1}) is surjective). Consequently, the application of Lemma 1.9.2 yields: Theorem 1.9.3. Let Q: C^{∞}(V_{1}) → C^{∞}(V_{2}) be an elliptic differential operator of order d > 0. Let Q^{∗}: C^{∞}(V_{2}) → C^{∞}(V_{1}) be the formal adjoint and define P = Q⊕Q^{∗}: C^{∞}(V_{1} ⊕V_{2}) → C^{∞}(V_{1} ⊕V_{2}). Let B = B_{1} ⊕B_{2} be a boundary condition such that (P, B) is elliptic with respect to the cone C − R_{+} − R_{−} and so that P_{B} is self-adjoint. Define S = +1 on V_{1} and −1 on V_{2} then: (a) index(Q_{B1} ) = Tr(S e^{−tP^{2} B}) for all t. (b) There exist local invariants a_{n}(x, Q) and a_{n}(y, Q, B_{1}) such that: Tr(S e^{−tP^{2} B}) ∼ _{∞ ∑ n=0} t^{^{n−m} 2d} ∫_{M} a_{n}(x, Q) dvol(x) +_{∞ ∑ n=0} t^{^{n−m+1} 2d} ∫_{dM} a_{n}(y, Q, B_{1}) dvol(y). (c)∫_{M} a_{n}(x, Q) dvol(x) + ∫_{dM} a_{n−1}(y, Q, B_{1}) dvol(y) = { index(Q_{B1} ) if n = m 0 if n ≠ m. This gives a local formula for the index; there is an analogue of Lemma 1.7.5 giving various functorial properties of these invariants we will discuss in the fourth chapter where we shall discuss the de Rham complex and the Gauss-Bonnet theorem for manifolds with boundary. We now specialize henceforth to first order operators. We decompose p(x, ξ) = _{∑ j} e_{j}(x)ξj and near the boundary express: p(y, 0, ζ, z) = e^{0}(y) · z +_{m−1 ∑ j=1} e^{j}(y)ζ^{j} = e^{0}(y) · z + p(y, 0, ζ, 0).

We study the ordinary differential equation: {ip_{0}∂/∂r + p(y, 0, ζ, 0)}f(r) = λf(r) or equivalently: −ip_{0}{∂/∂r + ip_{−1 0}p(y, 0, ζ, 0) − ip_{−1 0}λ)}f(r) = 0. With this equation in mind, we define: τ(y, ζ, λ) = ip_{−1 0}(p(y, 0, ζ, 0) − λ). Lemma 1.9.4. Let p(x, ξ) be self-adjoint and elliptic. Then τ(y, ζ, λ) has no purely imaginary eigenvalues for (ζ, λ) ≠ (0, 0) ∈ T^{∗}(dM) × (C − R_{+} − R_{−}). Proof: We suppose the contrary and set τ(y, ζ, λ)v = izv where z is real and v ≠ 0. This implies that: (p(y, 0, ζ, 0) − λ)v = p_{0}zv or equivalently that:  p(y, 0, ζ, −z)v = λv. Since p is self-adjoint, this implies λ = 0 so ζ ≠ 0 which contradicts the ellipticity of p. We define bundles Π_{±}(τ) over T^{∗}(dM)×{C−R_{+} −R_{−}}−(0, 0) to be the span of the generalized eigenvectors of τ which correspond to eigenvalues with positive/negative real part; Π_{+} ⊕ Π_{−} = V. The differential equation has the form:  {∂/∂r + τ}f = 0 so the condition lim_{r→∞} f(r) = 0 implies that f(0) ∈ Π_{+}(τ). Thus Π_{+}(τ) is the bundle of Cauchy data corresponding to solutions to this ODE. Since d = 1, the boundary condition B is just an endomorphism: B: V^{|dM} → W ′ and we conclude: Lemma 1.9.5. Let P be a first order formally self-adjoint elliptic differ- ential operator. Let B be a 0^{th} order boundary condition. Define τ(y, ζ, λ) = ip_{−1 0}(p(y, 0, ζ, 0) − λ). Then τ has no purely imaginary eigenvalues and we define Π_{±}(τ) to be bundles of generalized eigenvectors corresponding to eigenvalues with pos- itive real and negative real parts. (P, B) is elliptic with respect to the cone

Value Problems 77

C − R_{+} − R_{−} if and only if B: Π_{+}(τ) → W′ is an isomorphism for all (ζ, λ) ≠ (0, 0) ∈ T^{∗}(dM) × \{C − R_{+} − R_{−}\}. P_{B} is self-adjoint if and only if p_{0} N(B) is perpendicular to N(B). Proof: We have checked everything except the condition that P_{B} be self- adjoint. Since P is formally self-adjoint, it is immediate that: (P f, g) − (f, P g) = ∫_{dM} (−ip_{0}f, g). We know that on the boundary, both f and g have values in N(B) since they satisfy the boundary condition. Thus this vanishes identically if and only if (p_{0}f, g) = 0 for all f, g ∈ N(B) which completes the proof. We emphasize that these boundary conditions are much stronger than those required in the Atiyah-Bott theorem to define the index. They are much more rigid and avoid some of the pathologies which can occur other- wise. Such boundary conditions do not necessarily exist in general as we shall see later. We shall discuss the general case in more detail in Chapter 4. We com- plete this section by discussing the one-dimensional case case to illustrate the ideas involved. We consider V = [0, 1] × C^{2} and let P be the operator: P = −i ∂ ∂r ( {0 1 \atop 1 0} ) . At the point 0, we have: τ(λ) = −iλ ( {0 1 \atop 1 0} ) . Thus if Im(λ) > 0 we have Π_{+}(τ) = \{(_{a a})\} while if Im(λ) < 0 we have Π_{+}(τ) = \{(_{a −a})\}. We let B be the boundary projection which projects on the first factor—N(B) = \{(_{0 a})\}; we take the same boundary condition at x = 1 to define an elliptic self-adjoint operator P^{B}. Let P^{2} = − ∂2 ∂x2 with boundary conditions: Dirichlet boundary conditions on the first factor and Neumann boundary conditions on the second factor. The index of the problem is −1.

1.10. Eta and Zeta Functions. We have chosen to work with the heat equation for various technical reasons. However, much of the development of the subject has centered on the zeta function so we shall briefly indicate the relationship between these two in this section. We shall also define the eta invariant which plays an important role in the Atiyah-Singer index theorem for manifolds with boundary. Recall that Γ is defined by: Γ(s) = ∫_{∞ 0} t^{s−1}e^{−t}dt for Re(s) > 0. We use the functional equation sΓ(s) = Γ(s + 1) to ex- tend Γ to a meromorphic function on C with isolated simple poles at s = 0, −1, −2,…. Let P: C^{∞}(V) → C^{∞}(V) be an elliptic self-adjoint partial differential operator of order d > 0 with positive definite leading symbol. We showed in section 1.6 that this implies spec(P) is contained in [−C, ∞) for some constant C. We now assume that P itself is positive definite—i.e., spec(P) is contained in [ε, ∞) for some ε > 0. Proceeding formally, we define P^{s} by: P^{s} = 1 2πi ∫_{α} λ^{s}(P − λ)^{−1} dλ where α is a suitable path in the half-plane Re(λ) > 0. The estimates of section 1.6 imply this is smoothing operator if Re(s) ≪ 0 with a kernel function given by: L(s, x, y) = _{∑ n} λ_{s n}ϕ_{n}(x) ⊗ ϕ̄n(y); this converges to define a C^{k} kernel if Re(s) < s_{0}(k). We use the Mellin transform to relate the zeta and heat kernels. ∫_{∞ 0} t^{s−1}e^{−λt}dt = λ^{−s} ∫_{∞ 0}(λt)^{s−1}e^{−λt}d(λt) = λ^{−s}Γ(s). This implies that: Γ(s) Tr(P^{−s}) = ∫_{∞ 0} t^{s−1} Tr e^{−tP} dt. We define ζ(s, P) = Tr(P^{−s}); this is holomorphic for Re(s) ≫ 0. We decompose this integral into ∫_{1 0} + ∫_{∞ 1}. We have bounded the eigenvalues of

Eta and Zeta Functions 79

P away from zero. Using the growth estimates of section 1.6 it is immediate that: ∫_{∞ 1} t^{s−1} Tr e^{−tP} dt = r(s, P) defines an entire function of s. If 0 < t < 1 we use the results of section 1.7 to expand ^{Tr e−tP} = ^{∑ n≤n0} t^{^{n−m} d} a^{n}(P) + O (t^{^{n_{0} −m} d} ) a_{n}(P) = ∫_{M} a_{n}(x, P) dvol(x) where a_{n}(x, P) is a local scalar invariant of the jets of the total symbol of P given by Lemma 1.7.4. If we integrate the error term which is O (t^{^{n_{0} −m} d} ) from 0 to 1, we define a holomorphic function of s for Re(s)+_{n_{0}−m d} > 0. We integrate t^{s−1}t^{(n−m)} d between 0 and 1 to conclude: Γ(s) Tr(P^{−s}) = Γ(s)ζ(s, P) = ^{∑ n<n0} d(sd + n − m)^{−1}a^{n}(P) + rn0 where r_{n0} is holomorphic for Re(s) > −_{(n_{0}−m) d}. This proves Γ(s)ζ(s, P) extends to a meromorphic function on C with isolated simple poles. Fur- thermore, the residue at these poles is given by a local formula. Since Γ has isolated simple poles at s = 0, −1, −2,… we conclude that: Lemma 1.10.1. Let P be a self-adjoint, positive, elliptic partial differ- ential operator of order d > 0 with positive definite symbol. We define: ζ(s, P) = Tr(P^{−s}) = Γ(s)^{−1} ∫_{∞ 0} t^{s−1} Tr e^{−tP} dt. This is well defined and holomorphic for Re(s) ≫ 0. It has a meromor- phic extension to C with isolated simple poles at s = (m − n)/d for n = 0, 1, 2,…. The residue of ζ at these local poles is a_{n}(P)Γ((m − n)/d)^{−1}. If s = 0, −1,… is a non-positive integer, then ζ(s, P) is regular at this value of s and its value is given by a_{n}(P) Res_{s=(m−n)/d} Γ(s). a_{n}(P) is the invariant given in the asymptotic expansion of the heat equation. a_{n}(P) = ∫_{M} a_{n}(x, P) dvol(x) where a_{n}(x, P) is a local invariant of the jets of the total symbol of P. a_{n} vanishes if n is odd. Remark: If A is an auxilary differential operator of order a, we can define ζ(s, A, P) = Tr(AP^{−s}) = Γ(s)^{−1} ∫_{∞ 0} t^{s−1} Tr(Ae^{−tP}) dt.

80 1.10. Eta and Zeta Functions We apply Lemma 1.7.7 to see this has a meromorphic extension to C with isolated simple poles at s = (m + a − n)/d. The residue at these poles is given by the generalized invariants of the heat equation a_{n}(A, P) Γ{(m + a − n)/d}. There is a similar theorem if dM ≠ ∅ and if B is an elliptic boundary condition as discussed in section 1.9. If P is only positive semi-definite so it has a finite dimensional space corresponding to the eigenvalue 0, we define: ζ(s, P) = _{∑ λ_{n}>0} λ^{−s} n and Lemma 1.10 extends to this case as well with suitable modifications. For example, if M = S^{1} is the unit circle and if P = −∂^{2}/∂θ^{2} is the Laplacian, then: ζ(s, P) = 2^{∑ n>0} n−2s is essentially just the Riemann zeta function. This has a simple isolated pole at s = 1/2. If Q: C^{∞}(V_{1}) → C^{∞}(V_{2}), then the cancellation lemma of section 1.6 implies that: ε^{−s} index(Q) = ζ(s, Q^{∗}Q + ε) − ζ(s, QQ^{∗} + ε) for any ε > 0 and for any s. ζ is regular at s = 0 and its value is given by a local formula. This gives a local formula for index(Q). Using Lemma 1.10.1 and some functorial properties of the heat equation, it is not difficult to show this local formula is equivalent to the local formula given by the heat equation so no new information has resulted. The asymptotics of the heat equation are related to the values and residues of the zeta function by Lemma 1.10.1. If we do not assume that P is positive definite, it is possible to define a more subtle invariant which measures the difference between positive and negative spectrum. Let P be a self-adjoint elliptic partial differential operator of order d > 0 which is not necessarily positive definite. We define the eta invariant by: η(s, P) = ^{∑ λ_{n}>0} λ^{−s n} −^{∑ λ_{n}<0}(−λ^{n})^{−s} = ^{∑ λ_{n}≠0} sign(λ^{n})|λ^{n}|−s = Tr(P · (P^{2})^{−(s+1)/2} ).

Again, this is absolutely convergent and defines a holomorphic function if Re(s) ≫ 0. We can also discuss the eta invariant using the heat equation. The identity: ∫^{∞ 0} t^{(s−1)/2} λe^{−λ^{2}t} dt = Γ((s + 1)/2) sign(λ)|λ|−s implies that: η(s, P) = Γ((s + 1)/2)^{−1} ∫_{∞ 0} t^{(s−1)/2} Tr{P e^{−tP 2} } dt. Again, the asymptotic behavior at t = 0 is all that counts in producing poles since this decays exponentially at ∞ assuming P has no zero eigenvector; if dim N(P) > 0 a seperate argument must be made to take care of this eigenspace. This can be done by replacing P by P + ε and letting ε → 0. Lemma 1.7.7 shows that there is an asymptotic series of the form: Tr(P e^{−tP 2} ) ∼ _{∞ ∑ n=0} t^{^{n−m−d} 2d} a_{n}(P, P^{2}) for  a_{n}(P, P^{2}) = ∫_{M} a_{n}(x, P, P^{2}) dvol(x). This is a local invariant of the jets of the total symbol of P. We substitute this asymptotic expansion into the expression for η to see: η(s, P)Γ((s + 1)/2)^{−1} = ^{∑ n≤n0}  2d ds + n − ma^{n}(P, P^{2}) + rn0 where r_{n0} is holomorphic on a suitable half-plane. This proves η has a suitable meromorphic extension to C with locally computable residues. Unfortunately, while it was clear from the analysis that ζ was regular at s = 0, it is not immediate that η is regular at s = 0 since Γ does not have a pole at _{1 2} to cancel the pole which may be introduced when n = m. in fact, if one works with the local invariants involved, a_{n}(x, P, P^{2}) ≠ 0 in general so the local poles are in fact present at s = 0. However, it is a fact which we shall discuss later that η is regular at s = 0 and we will definẽ η(p) = 1 2{η(0, p) + dim N(p)} mod Z (we reduce modulo Z since η has jumps in 2Z as eigenvalues cross the origin).

We compute a specific example to illustrate the role of η in measuring spectral asymmetry. Let P = −i∂/∂θ on C^{∞}(S^{1}), then the eigenvalues of p are the integers so η(s, p) = 0 since the spectrum is symmetric about the origin. Let a ∈ R and define: P_{a} = P − a,  η(s, P_{a}) = _{∑ n∈Z} sign{n − a}|n − a|^{−s}. We differentiate this with respect to the parameter a to conclude: d daη(s, P (a)) = _{∑ n∈Z} s((n − a)^{2})^{−(s+1)/2} . If we compare this with the Riemann zeta function, then^{∑ n∈Z((n − a)2)−(s+1)/2} has a simple pole at s = 0 with residue 2 and therefore: d daη(s, P (a))||||_{s=0} = 2. Since η vanishes when a = 0, we integrate this with respect to a to conclude: η(0, P (a)) = 2a + 1 mod 2Z  and η̃(P (a)) = a + 1 2 is non-trivial. (We must work modulo Z since spec P (a) is periodic with period 1 in a). We used the identity: d daη(s, P_{a}) = −s Tr(Ṗa(P_{2 a})^{−(s+1)/2} ) in the previous computation; it in fact holds true in general: lemma 1.10.2. Let P (a) be a smooth 1-parameter family of elliptic self- adjoint partial differential operators of order d > 0. Assume P (a) has no zero spectrum for a in the parameter range. Then if “·” denotes differen- tiation with respect to the parameter a, η̇(s, P (a)) = −s Tr(Ṗ (a)(P (a)^{2})^{−(s+1)/2} ). If P (a) has zero spectrum, we regard η(s, P (a)) ∈ C/Z. Proof: If we replace P by P^{k} for an odd positive integer k, then η(s, P^{k}) = η(ks, P). This shows that it suffices to prove Lemma 1.10.2 for d ≫ 0.

By Lemma 1.6.6, (P − λ)^{−1} is smoothing and hence of trace class for d large. We can take trace inside the integral to compute: η(s, P (a)) = 1 2πi {∫_{γ1} λ^{−s} Tr((P (a) − λ)^{−1}) dλ − ∫^{γ2} (−λ)^{−s} Tr((P (a) − λ)^{−1} dλ} where γ_{1} and γ_{2} are paths about the positive/negative real axis which are oriented suitably. We differentiate with respect to the parameter a to express: d da((P (a) − λ)^{−1} = −(P (a) − λ)^{−1}Ṗ (a)(P (a) − λ)^{−1}. Since the operators involved are of Trace class, Tr ( d da(P (a) − λ)^{−1}) = − Tr(Ṗ (a)(P (a) − λ)^{−2}). We use this identity and bring trace back outside the integral to compute: η(s, P (a)) = 1 2πi Tr(Ṗ (a)){ ∫_{γ1} −λ^{−s}(P (a) − λ)^{−2} dλ + ∫_{γ2} (−λ)^{−s}(P (a) − λ)^{−2} dλ}. We now use the identity: d dλ((P (a) − λ)^{−1}) = (P (a) − λ)−2 to integrate by parts in λ in this expression. This leads immediately to the desired formula. We could also calculate using the heat equation. We proceed formally: dη da (s, P (a)) = Γ{(s + 1)/2}^{−1} ∫_{∞ 0} t^{(s−1)/2} Tr ( d da(P (a)e^{−tP (a)2} )) dt = Γ{(s + 1)/2}^{−1} ∫_{∞ 0} t^{(s−1)/2} Tr (dP da (1 − 2tP^{2})e^{−tP 2} ) dt = Γ{(s + 1)/2}^{−1} ∫_{∞ 0} t^{(s−1)/2} (1 + 2t d dt) Tr (dP da e^{−tP 2} ) dt. We now integrate by parts to compute:

84 1.10. Eta and Zeta Functions = Γ{(s + 1)/2}^{−1} ∫_{∞ 0} −st^{(s−1)/2} Tr (dP da e^{−tP 2} ) dt ∼ −sΓ{(s + 1)/2}^{−1} _{∑ n} ∫_{1 0} t^{(s−1)/2} t^{(n−m−d)/2d} a_{n} (dP da , P^{2}) dt ∼ −sΓ{(s + 1)/2}^{−1} _{∑ n} 2/(s + (n − m)/d)a_{n} (dP da , P^{2}) . In particular, this shows that dη̃ da is regular at s = 0 and the value −Γ(_{1 2} )^{−1} a^{m} (dP da , P^{2}) is given by a local formula. The interchange of order involved in using global trace is an essential part of this argument. Tr(Ṗ (a)(P^{2})^{−(s+1)/2} ) has a meromorphic extension to C with isolated simple poles. The pole at s = −1/2 can be present, but is cancelled off by the factor of −s which multiplies the expression. Thus d da Res_{s=0} η(s, P (a)) = 0. This shows the global residue is a homotopy invariant; this fact will be used in Chapter 4 to show that in fact η is regular at the origin. This argument does not go through locally, and in fact it is possible to construct operators in any dimension m ≥ 2 so that the local eta function is not regular at s = 0. We have assumed that P (a) has no zero spectrum. If we supress a finite number of eigenvalues which may cross the origin, this makes no contribution to the residue at s = 0. Furthermore, the value at s = 0 changes by jumps of an even integer as eigenvalues cross the origin. This shows η̃ is in fact smooth in the parameter a and proves: Lemma 1.10.3. (a) Let P be a self-adjoint elliptic partial differential operator of positive order d. Define: η(s, P) = Tr(P (P^{2})^{−(s+1)/2} ) = Γ{(s+1)/2}^{−1} ∫_{∞ 0} t^{(s−1)/2} Tr(P e^{−tP 2} ) dt. This admits a meromorphic extension to C with isolated simple poles at s = (m − n)/d. The residue of η at such a pole is computed by: Res_{s=(m−n)/d} = 2Γ{(m + d − n)/2d}^{−1} ∫_{M} a_{n}(x, P, P^{2}) dvol(x) where a_{n}(x, P, P^{2}) is defined in Lemma 1.7.7; it is a local invariant in the jets of the symbol of P.

Eta and Zeta Functions 85

(b) Let P (a) be a smooth 1-parameter family of such operators. If we assume eigenvalues do not cross the origin, then: d daη(s, P (a)) = −s Tr (dP da · (P (a)^{2})^{−(s+1)/2} ) = −2sΓ{(s + 1)/2}^{−1} ∫_{∞ 0} t^{(s−1)/2} Tr (dP da · e^{−tP 2} ) dt. Regardless of whether or not eigenvalues cross the origin, η̃(P (a)) is smooth in R mod Z in the parameter a and d daη̃(P (a)) = −Γ(^{1 2} )^{−1}a^{m} (dP da , P^{2}) = −Γ(_{1 2} )^{−1} ∫_{M} a_{m} (x, dP da , P^{2}) dvol(x) is the local invariant in the jets of the symbols of (dP da , P^{2}) given by Lemma 1.7.7. In the example on the circle, the operator P (a) is locally isomorphic to P. Thus the value of η at the origin is not given by a local formula. This is a global invariant of the operator; only the derivative is locally given. It is not necessary to assume that P is a differential operator to define the eta invariant. If P is an elliptic self-adjoint pseudo-differential operator of order d > 0, then the sum defining η(s, P) converges absolutely for s ≫ 0 to define a holomorphic function. This admits a meromorphic extension to the half-plane Re(s) > −δ for some δ > 0 and the results of Lemma 1.10.3 continue to apply. This requires much more delicate estimates than we have developed and we shall omit details. The reader is referred to the papers of Seeley for the proofs. We also remark that it is not necessary to assume that P is self-adjoint; it suffices to assume that det(p(x, ξ)−it) ≠ 0 for (ξ, t) ≠ (0, 0) ∈ T^{∗}M ×R. Under this ellipticity hypothesis, the spectrum of P is discrete and only a finite number of eigenvalues lie on or near the imaginary axis. We define: η(s, P) = ^{∑ Re(λ_{i})>0}(λ^{i})^{−s} −^{∑ Re(λ_{i})<0}(−λ^{i})−s̃ η(P) = 1 2 {η(s, P) − 1s Res_{s=0} η(s, P) + dim N(iR)}_{s=0} mod Z where dim N(iR) is the dimension of the finite dimensional vector space of generalized eigenvectors of P corresponding to purely imaginary eigenval- ues.

In section 4.3, we will discuss the eta invariant in further detail and use it to define an index with coefficients in a locally flat bundle using secondary characteristic classes. If the leading symbol of P is positive definite, the asymptotics of Tr(e^{−tP}) as t → 0^{+} are given by local formulas integrated over the manifold. Let A(x) = a_{0}x^{a}+ ⋯+a_{a} and B(x) = b_{0}x^{b}+ ⋯+b_{b} be polynomials where b_{0} > 0. The operator A(P)e^{−tB(P)} is infinitely smoothing. The asymptotics of Tr(A(P)e^{−tB(P)} ) are linear combinations of the invariants a_{N} (P) giving the asymptotics of Tr(e^{−tP}). Thus there is no new information which results by considering more general operators of heat equation type. If the leading symbol of P is indefinite, one must consider both the zeta and eta function or equivalently: Tr(e^{−tP 2} )  and  Tr(P e^{−tP 2} ). Then if b_{0} > 0 is even, we obtain enough invariants to calculate the asymp- totics of Tr(A(P)e^{−tB(P)} ). We refer to Fegan-Gilkey for further details.