j. differential geometry 62 (2002) 17-38

SHARP AFFINE L_{P} SOBOLEV INEQUALITIES

ERWIN LUTWAK, DEANE YANG & GAOYONG ZHANG

Abstract

A sharp affine L_{p} Sobolev inequality for functions on Euclidean n-space is established. This new inequality is significantly stronger than (and directly implies) the classical sharp L_{p} Sobolev inequality of Aubin and Talenti, even though it uses only the vector space structure and standard Lebesgue measure on ℝ^{n}. For the new inequality, no inner product, norm, or conformal structure is needed; the inequality is invariant under all affine transformations of ℝ^{n}.

0. Introduction

In this paper we prove a sharp affine L_{p} Sobolev inequality for functions on ℝ^{n}. The new inequality is significantly stronger than (and directly implies) the classical sharp L_{p} Sobolev inequality of Aubin [2] and Talenti [39], even though it uses only the vector space structure and standard Lebesgue measure on ℝ^{n}. For the new inequality, no inner product, norm, or conformal structure is needed at all. In other words, the inequality is invariant under all affine transformations of ℝ^{n}. That such an inequality exists is surprising because the classical sharp L_{p} Sobolev inequality relies strongly on the Euclidean geometric structure of ℝ^{n}, especially on the isoperimetric inequality.

Zhang [42] formulated and proved the sharp affine L_{1} Sobolev inequality and established its equivalence to an L_{1} affine isoperimetric inequality that is also proved in [42]. He also showed that the affine L_{1} Sobolev inequality is stronger than the classical L_{1} Sobolev inequality.

The L_{1} Sobolev inequality is known to be equivalent to the isoperimetric inequality (see, for example, [17], [18], [34], [10], [35], and [38]).

Research supported, in part, by NSF Grant DMS–0104363.

Received 03/30/2002.

The geometry behind the sharp L_{p} Sobolev inequality is also the isoperimetric inequality. For the affine Sobolev inequalities the situation is quite different. The geometric inequality and the critical tools used to establish the affine L_{1} Sobolev inequality are not strong enough to enable us to establish the affine L_{p} Sobolev inequality for p > 1. A new geometric inequality and new tools are needed. The inequality needed is an affine L_{p} affine isoperimetric inequality recently established by the authors in [31] (see Campi and Gronchi [11] for a recent alternate approach). We will also need the solution of an L_{p} extension of the classical Minkowski problem obtained in [29]. It is crucial to observe that while the geometric core of the classical L_{p} Sobolev inequality (i.e., the isoperimetric inequality) is the same for all p, the geometric inequality (i.e., the affine L_{p} isoperimetric inequality) behind the new affine L_{p} Sobolev inequality is different for different p.

Let ℝ^{n} denote n–dimensional Euclidean space; throughout we will assume that n ≥ 2. Let H^{1,p}(ℝ^{n}) denote the usual Sobolev space of real-valued functions of ℝ^{n} with L_{p} partial derivatives.

The classical sharp L_{p} Sobolev inequality of Aubin [2] and Talenti [39] states that if f ∈ H^{1,p}(ℝ^{n}), with real p satisfying 1 < p < n, and if q is given by \frac{1}{q} = \frac{1}{p} − \frac{1}{n}, then

(∫_{ℝ^{n}} |∇f|^{p} dx)^{\frac{1}{p}} ≥ c_{0} ∥f∥_{q}, (0.1)

where |∇f| is the Euclidean norm of the gradient of f, while ∥f∥_{q} is the usual L_{q} norm of f in ℝ^{n}, and

c_{0} = n^{\frac{1}{p}} (\frac{n − p}{p − 1})^{1− \frac{1}{p}} [ω_{n}Γ (\frac{n}{p}) Γ (n + 1 − \frac{n}{p}) /Γ(n)]^{\frac{1}{n}} ,

where ω_{n} is the n-dimensional volume enclosed by the unit sphere S^{n−1} in ℝ^{n}. Generalizations of (0.1) and related problems have been much studied (see, e.g., [3], [5], [6], [7], [8], [9], [12], [13], [16], [23], [24], [26], [27], [41], [42], and the references therein).

Since the case p = 1 of the sharp affine L_{p} Sobolev inequality was settled in [42], in this paper we will focus exclusively on the case p > 1.

sharp affine L_{p} sobolev inequalities 19

The basic concept behind our new inequality is a Banach space that we will associate with each function in H^{1,p}(ℝ^{n}). The critical observation here is that this association is affine in nature. For real p ≥ 1, we associate with each f ∈ H^{1,p}(ℝ^{n}) a Banach norm ∥ · ∥_{f,p} on ℝ^{n}. For v ∈ S^{n−1} define

∥v∥_{f,p} = ∥D_{v}f∥_{p} = (∫_{ℝ^{n}} |D_{v}f(x)|^{p}dx)^{\frac{1}{p}} = (∫_{ℝ^{n}} |v · ∇f(x)|^{p}dx)^{\frac{1}{p}} ,

where D_{v}f is the directional derivative of f in the direction v. The integral on the right immediately provides the extension of ∥ · ∥_{f,p} from S^{n−1} to ℝ^{n}. Now (ℝ^{n}, ∥ · ∥_{f,p}) is the n-dimensional Banach space that we shall associate with f. Its unit ball B_{p}(f) = {v ∈ ℝ^{n} : ∥v∥_{f,p} ≤ 1} is a symmetric convex body in ℝ^{n} and our new inequality states that the volume of this unit ball, |B_{p}(f)|, can be bounded from above by the reciprocal of the ordinary L_{q}-norm of f. Specifically, we have:

Theorem 1. Suppose p ∈ (1, n) and q is given by \frac{1}{q} = \frac{1}{p} − \frac{1}{n}. If f ∈ H^{1,p}(ℝ^{n}), then

|B_{p}(f)|^{1/n} ≤ c_{1}/∥f∥_{q}, (0.2)

where the best possible c_{1} is given by

c_{1} = (\frac{p − 1}{n − p})^{1− \frac{1}{p}} (\frac{Γ(n)}{Γ(\frac{n}{p})Γ(n + 1 − \frac{n}{p})})^{\frac{1}{n}} (\frac{\sqrt{π}Γ(\frac{n+p}{2})}{nΓ(\frac{n}{2})Γ(\frac{p+1}{2})})^{\frac{1}{p}}

and equality is attained when

f(x) = (a + |A(x − x_{0})|^{\frac{p}{p−1}} )^{1− \frac{n}{p}} ,

with A ∈ GL(n), real a > 0, and x_{0} ∈ ℝ^{n}.

Since the volume of the symmetric convex body B_{p}(f) is obviously given by

|B_{p}(f)| = \frac{1}{n} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv,

we can rewrite our main theorem as the following affine L_{p} Sobolev inequality:

Theorem 1′. Suppose p ∈ (1, n) and q is given by \frac{1}{q} = \frac{1}{p} − \frac{1}{n}. If f ∈ H^{1,p}(ℝ^{n}), then

(∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{−1/n} ≥ c_{2} ∥f∥_{q}, (0.3) where the best possible c_{2} is given by

c_{2} = (\frac{n − p}{p − 1})^{1− \frac{1}{p}} (\frac{Γ(\frac{n}{p})Γ(n + 1 − \frac{n}{p})}{Γ(n + 1)})^{\frac{1}{n}} (\frac{nΓ(\frac{n}{2})Γ(\frac{p+1}{2})}{\sqrt{π}Γ(\frac{n+p}{2})})^{\frac{1}{p}}

and equality is attained when

f(x) = (a + |A(x − x_{0})|^{\frac{p}{p−1}} )^{1− \frac{n}{p}} ,

with A ∈ GL(n), real a > 0, and x_{0} ∈ ℝ^{n}.

Using the obvious fact that

\frac{1}{n!} ∫_{ℝ^{n}} e^{−∥D_{v}f∥_{p}}dv = \frac{1}{n} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv,

we can in turn rewrite Theorem 1′ as:

Theorem 1′′. Suppose p ∈ (1, n) and q is given by \frac{1}{q} = \frac{1}{p} − \frac{1}{n}. If f ∈ H^{1,p}(ℝ^{n}), then

(∫_{ℝ^{n}} e^{−∥D_{v}f∥_{p}}dv)^{− \frac{1}{n}} ≥ c_{3} ∥f∥_{q}, (0.4)

where the best possible c_{3} is given by

c_{3} = (\frac{n − p}{p − 1})^{1− \frac{1}{p}} (\frac{Γ(\frac{n}{p})Γ(n + 1 − \frac{n}{p})}{Γ(n)Γ(n + 1)})^{\frac{1}{n}} (\frac{nΓ(\frac{n}{2})Γ(\frac{p+1}{2})}{\sqrt{π}Γ(\frac{n+p}{2})})^{\frac{1}{p}} ,

and equality is attained when

f(x) = (a + |A(x − x_{0})|^{\frac{p}{p−1}} )^{1− \frac{n}{p}} ,

with A ∈ GL(n), real a > 0, and x_{0} ∈ ℝ^{n}.

Observe that inequality (0.4), and thus also inequality (0.3), is invariant under affine transformations of ℝ^{n}, while the L_{p} Sobolev inequality (0.1) is invariant only under rigid motions.

That the affine L_{p} Sobolev inequality (0.3) or (0.4) is stronger than the classical L_{p} Sobolev inequality (0.1) follows directly from the Hölder inequality, as will be shown in Section 7. We also note that the affine L_{2} Sobolev inequality and the classical L_{2} Sobolev inequality are equivalent under an affine transformation since the L_{2} Banach norm ∥ · ∥_{f,2} is Euclidean.

In Section 8, we present an application of the affine L_{p} Sobolev inequality to information theory. For a random vector X in a finite dimensional Banach space that is associated to a function f, we prove a sharp inequality that gives the best lower bound of the moments of X with respect to the Banach norm in terms of the λ-Renýi entropy of X and the L_{q} norm of f. Additional applications will be given in a forthcoming paper.

1. Background

For quick reference we list some facts about convex bodies. See [19], [37] and [40] for additional details. A convex body is a compact convex set in ℝ^{n} with nonempty interior. In this paper it will always be assumed that a convex body contains the origin in its interior. A convex body K is uniquely determined by its support function h(K, · ) = h_{K} : ℝ^{n} → [0, ∞), defined for v ∈ ℝ^{n} by

h_{K}(v) = max{v · x : x ∈ K},

where v · x denotes the usual inner product of v and x in ℝ^{n}. The n-dimensional volume of K will be denoted by V (K) or |K|.

For real p ≥ 1, convex bodies K, L and real ε > 0, the Minkowski-Firey L_{p} combination, K +_{p} ε · L, is the convex body whose support function is given

h(K +_{p} ε·L, ·)^{p} = h(K, ·)^{p} + εh(L, · )^{p}.

The L_{p}-mixed volume V_{p}(K, L) of convex bodies K and L is defined by

V_{p}(K, L) = \frac{p}{n} lim_{ε→0^{+}} \frac{V (K +_{p} ε · L) − V (K)}{ε}.

Note that V_{p}(Q, Q) = V (Q) for each convex body Q. It was shown in [29] that there exists a unique finite positive Borel measure S_{p}(K, · ) on S^{n−1} such that

V_{p}(K, Q) = \frac{1}{n} ∫_{S^{n−1}} h_{Q}(v)^{p}dS_{p}(K, v), (1.1)

for each convex body Q. The measure S_{p}(K, · ) is called the L_{p}-surface area measure of K. The measure S_{1}(K, · ) = S_{K} is the classical surface area measure of K. It was shown in [29] that the measure S_{p}(K, · ) is absolutely continuous with respect to S_{K} and the Radon-Nikodym derivative

\frac{dS_{p}(K, · )}{dS_{K}} = h_{K}^{1−p}.

If the boundary, ∂K, of K is C^{2} with positive curvature, then the Radon-Nikodym derivative of S_{K} with respect to the Lebesgue measure on S^{n−1} is the reciprocal of the Gauss curvature of ∂K (when viewed as a function of the outer normals of ∂K).

A compact domain is the closure of a bounded open set. For compact domains M_{1}, M_{2} and real λ_{1}, λ_{1} ≥ 0, the Minkowski linear combination λ_{1}M_{1} + λ_{2}M_{2} is defined by

λ_{1}M_{1} + λ_{2}M_{2} = {λ_{1}x_{1} + λ_{2}x_{2} : x_{1} ∈ M_{1} and x_{2} ∈ M_{2}}.

The Brunn-Minkowski inequality states that if M_{1}, M_{2} are compact domains in ℝ^{n} and λ_{1}, λ_{1} ≥ 0, then

V (λ_{1}M_{1} + λ_{2}M_{2})^{1/n} ≥ λ_{1}V (M_{1})^{1/n} + λ_{2}V (M_{2})^{1/n},

where V denotes n-dimensional Lebesgue measure. If M is a compact domain and K is a convex body in ℝ^{n}, define the mixed volume, V_{1}(M, K), of M and K by

nV_{1}(M, K) = lim inf_{ε→0^{+}} \frac{V (M + εK) − V (M)}{ε}.

We shall require the following Minkowski mixed volume inequality for compact domains: If M is a compact domain in ℝ^{n} and K is a convex body in ℝ^{n}, then

V_{1}(M, K)^{n} ≥ V (M)^{n−1}V (K). (1.2)

Note that (1.2) follows immediately from the definition of mixed volumes and the Brunn-Minkowski inequality:

V_{1}(M, K) = \frac{1}{n} lim inf_{ε→0^{+}} \frac{V (M + εK) − V (M)}{ε}

≥ \frac{1}{n} lim_{ε→0^{+}} \frac{[V (M)^{1/n} + εV (K)^{1/n}]^{n} − V (M)}{ε}

= V (M)^{\frac{n−1}{n}} V (K)^{\frac{1}{n}} .

We will also require the following integral representation: Suppose M is a compact domain with C^{1} boundary, ∂M. Then, if K is a convex body in ℝ^{n},

V_{1}(M, K) = \frac{1}{n} ∫_{∂M} h_{K}(ν(x))dS_{M}(x), (1.3)

where ν(x) denotes the exterior unit normal at x ∈ ∂M, and dS_{M}(x) is the surface area element at x ∈ ∂M. Identity (1.3) can be found, e.g., in [42].

2. Affine L_{p} isoperimetric inequalities

We require an L_{p}-affine isoperimetric inequality that was first proved in [31] (see Campi and Gronchi [11] for an alternative proof and generalizations). This inequality is one of the key ingredients in the proof of Theorem 1. Special cases of this new inequality and their relations to other affine isoperimetric inequalities can be found in, e.g., [28] and [25].

While the L_{p} mixed volume V_{p}( · , · ) has been defined only for compact convex sets that contain the origin in their interiors, a simple continuity argument allows us to extend the definition of the L_{p}-mixed volume V_{p}(K, L) to the case where K is a compact convex set that contains the origin in its interior and L is a compact convex set that contains the origin in its relative interior. For u ∈ S^{n−1}, let u denote the line segment connecting the points −u/2 to u/2. Note that from (1.1) we have

V_{p}(K, u) = \frac{1}{2^{p}n} ∫_{S^{n−1}} |v · u|^{p}dS_{p}(K, v), (2.1)

for each u ∈ S^{n−1}.

Let c_{4} be defined by

c_{4} = \frac{\sqrt{π}Γ(\frac{n+p}{2})}{Γ(\frac{n}{2} + 1)Γ(\frac{p+1}{2})}.

The following affine isoperimetric inequality was established in [31] and will be critical in establishing the affine L_{p} Sobolev inequality:

Theorem 2.1. If p > 1 and K is a convex body in ℝ^{n} that contains the origin in its interior, then

(∫_{S^{n−1}} V_{p}(K, v)^{− \frac{n}{p}} dv)^{\frac{p}{n}} V (K)^{\frac{n−p}{n}} ≤ 2^{p−1}n^{1+ \frac{p}{n}} c_{4}, (2.2)

with equality if and only if K is an ellipsoid centered at the origin.

As an aside, we note that the actual inequality presented in [31] relates the volume of a convex body to that of its polar L_{p} projection body. However, the polar coordinate formula for volume quickly shows the equivalence of (2.2) and the polar L_{p} projection inequality that was established in [31].

3. The L_{p} Minkowski problem

We shall construct a family of convex bodies from a given function by using the solution to the even L_{p}-Minkowski problem. This will allow us to use the affine isoperimetric inequality (2.2) to establish Theorem 1.

A Borel measure on S^{n−1} is said to be even if for each Borel set ω ⊂ S^{n−1} the measure of ω and −ω = {−x : x ∈ ω} are equal. In [29], the following solution to the even case of the L_{p}-Minkowski problem is given:

Theorem 3.1. Suppose μ is an even positive measure on S^{n−1} that is not supported on a great hypersphere of S^{n−1}. Then for real p ≥ 1 such that p ≠ n there exists a unique origin-symmetric convex body K in ℝ^{n} whose L_{p}-surface area measure is μ; i.e.,

μ = S_{p}(K, · ).

We now define for functions (rather than bodies) the notions of L_{p} mixed volumes. Suppose f ∈ H^{1,p}(ℝ^{n}) ∩ C^{∞}(ℝ^{n}). For each real t > 0, define the level set,

[f]_{t} = {x ∈ ℝ^{n} : |f(x)| ≥ t}.

Then [f]_{t} is compact for each t > 0. By Sard’s theorem, for almost all t > 0, the boundary of the level set ∂[f]_{t} is a C^{1} submanifold with everywhere nonzero normal vector ∇f. Abbreviate the surface area element of ∂[f]_{t} by dS_{t}. If Q is a compact convex set that contains the origin in its relative interior, then define the L_{p}-mixed volume V_{p}(f, t, Q), by

V_{p}(f, t, Q) = \frac{1}{n} ∫_{∂[f]_{t}} h_{Q}(ν(x))^{p}|∇f(x)|^{p−1}dS_{t}(x), (3.1)

where ν(x) = ∇f(x)/|∇f(x)| is the outer unit normal at x ∈ ∂[f]_{t}. In particular, when Q is the line segment joining the points −v/2 and v/2, we have

V_{p}(f, t, v) = \frac{1}{2^{p}n} ∫_{∂[f]_{t}} |v · ∇f(x)|^{p}|∇f(x)|^{−1}dS_{t}(x), (3.2)

for each v ∈ S^{n−1}.

The following lemma shows that, for each fixed real p ≥ 1, there is a natural way to associate a family of convex bodies with a given function.

Lemma 3.2. If f ∈ H^{1,p}(ℝ^{n}) ∩ C^{∞}(ℝ^{n}), then for almost every t > 0, there exists an origin-symmetric convex body K_{t} whose volume is given by

V (K_{t}) = V_{p}(f, t, K_{t}) (3.3a)

and such that for all v ∈ S^{n−1},

V_{p}(f, t, v) = V_{p}(K_{t}, v). (3.3b)

Proof. Define the positive Borel measure μ_{t} on S^{n−1} by

∫_{S^{n−1}} g(v)dμ_{t}(v) = ∫_{∂[f]_{t}} g(ν(x))|∇f(x)|^{p−1}dS_{t}(x), (3.4)

for each continuous g : S^{n−1} → ℝ. Define the even Borel measure μ_{t}^{∗} on S^{n−1} by letting

μ_{t}^{∗}(ω) = \frac{1}{2}μ_{t}(ω) + \frac{1}{2}μ_{t}(−ω),

for each Borel ω ⊂ S^{n−1}. Obviously, for each continuous even g : S^{n−1} → ℝ,

∫_{S^{n−1}} g(v) dμ_{t}^{∗}(v) = ∫_{S^{n−1}} g(v) dμ_{t}(v). (3.5)

From (3.5), (3.4), and the fact that [f]_{t} has nonempty interior, it follows that for each u ∈ S^{n−1},

∫_{S^{n−1}} |u · v|dμ_{t}^{∗}(v) = ∫_{S^{n−1}} |u · v|dμ_{t}(v)
= ∫_{∂[f]_{t}} |u · ν(x)||∇f(x)|^{p−1}dS_{t}(x) > 0.

Hence, the measure μ_{t}^{∗} is not supported on any great hypersphere of S^{n−1}. By Theorem 3.1, there exists a unique origin-symmetric convex body K_{t} so that

dμ_{t}^{∗} = dS_{p}(K_{t}, · ) = h_{K_{t}}^{1−p}dS_{K_{t}}. (3.6)

To see that for each origin-symmetric convex body Q,

V_{p}(K_{t}, Q) = V_{p}(f, t, Q), (3.7)

note that from (1.1) and (3.6), (3.5), (3.4) and definition (3.1), it follows that

V_{p}(K_{t}, Q) = \frac{1}{n} ∫_{S^{n−1}} h_{Q}(v)^{p} dμ_{t}^{∗}(v)

= \frac{1}{n} ∫_{S^{n−1}} h_{Q}(v)^{p} dμ_{t}(v)

= \frac{1}{n} ∫_{∂[f]_{t}} h_{Q}(ν(x))^{p}|∇f(x)|^{p−1}dS_{t}(x)

= V_{p}(f, t, Q).

Now (3.7) and a continuity argument immediately yields (3.3b). To get (3.3a) take Q = K_{t} in (3.7) and recall that V_{p}(Q, Q) = V (Q). q.e.d.

4. An integral inequality

The following well-known (see, e.g., [1]) consequence of Bliss’ inequality [4] will be needed. For the sake of completeness we include an elementary proof that uses techniques similar to ones used in [15].

Lemma 4.1. Let f be a nonnegative differentiable function in (0, ∞), q = \frac{np}{n−p}, and 1 < p < n. If the integrals exist, then

(∫_{0}^{∞} |f′(x)|^{p}x^{n−1}dx)^{\frac{1}{p}} ≥ c_{5} (∫_{0}^{∞} f(x)^{q}x^{n−1}dx)^{\frac{1}{q}} ,

where

c_{5} = n^{\frac{1}{q}} (\frac{n − p}{p − 1})^{1− \frac{1}{p}} [Γ (\frac{n}{p}) Γ (n + 1 − \frac{n}{p}) /Γ(n)]^{\frac{1}{n}} = (nω_{n})^{− \frac{1}{n}} c_{0},

Equality holds if f(x) = (ax^{\frac{p}{p−1}} + b)^{1− \frac{n}{p}}, with a, b > 0.

Proof. It suffices to prove the inequality for a nonnegative compactly supported smooth function f satisfying

∫_{0}^{∞} f(x)^{q}x^{n−1}dx = 1.

Let f_{0} : (0, ∞) → [0, ∞) be a continuous function that is supported on a bounded interval [0, R) for some R > 0 and that satisfies

∫_{0}^{∞} f_{0}(x)^{q}x^{n−1}dx = ∫_{0}^{∞} f_{0}(x)^{q}x^{p^{∗}+n−1}dx = 1,

where p^{∗} = \frac{p}{p−1}. Define y : [0, ∞) → [0, R] by

∫_{0}^{x} f(s)^{q}s^{n−1}ds = ∫_{0}^{y(x)} f_{0}(t)^{q}t^{n−1}dt.

It follows that

(4.1)  f_{0}(y)^{q− \frac{q}{n}} y^{n−1}y′ = f(x)^{q− \frac{q}{n}} x^{n−1} [(\frac{y}{x})^{n−1} y′]^{\frac{1}{n}}

≤ \frac{1}{n}f(x)^{q− \frac{q}{n}} x^{n−1} ((n − 1)\frac{y}{x} + y′)

= \frac{1}{n}f(x)^{q− \frac{q}{n}} (x^{n−1}y)′.

Equality in the inequality holds if and only if y = λx, λ > 0. Integration by parts and the Hölder inequality give

∫_{0}^{∞} f(x)^{q− \frac{q}{n}} (x^{n−1}y)′dx (4.2)

= − (q − \frac{q}{n}) ∫_{0}^{∞} f(x)^{q− \frac{q}{n} −1}f′(x)x^{n−1}y dx

≤ (q − \frac{q}{n}) ∫_{0}^{∞} f(x)^{q− \frac{q}{n} −1}|f′(x)|x^{n−1}y dx

≤ (q − \frac{q}{n}) (∫_{0}^{∞} y^{p^{∗}}f^{q}x^{n−1}dx)^{\frac{1}{p^{∗}}} (∫_{0}^{∞} |f′|^{p}x^{n−1}dx)^{\frac{1}{p}}

= (q − \frac{q}{n}) (∫_{0}^{∞} f_{0}^{q}y^{p^{∗}+n−1}dy)^{\frac{1}{p^{∗}}} (∫_{0}^{∞} |f′|^{p}x^{n−1}dx)^{\frac{1}{p}} .

By (4.1) and (4.2),

(∫_{0}^{∞} |f′|^{p}x^{n−1}dx)^{\frac{1}{p}} ≥ \frac{n(n − p)}{p(n − 1)} ∫_{0}^{∞} f_{0}^{q−q/n}y^{n−1}dy. (4.3)

Since (4.3) holds for any compactly supported function f_{0}, it holds for any positive continuous function. Moreover, equality holds for (4.3) if y = λx and f′ = βf^{\frac{n}{n−p}} y^{\frac{1}{p−1}} for some constant β. Integrating this gives the extremal function. In particular, the desired inequality follows by setting f_{0}(y) = (ay^{\frac{p}{p−1}} + b)^{1− \frac{n}{p}}, where a and b are chosen so that f_{0} satisfies the required normalizations. q.e.d.

5. A lemma about rearrangements

For f : ℝ^{n} → ℝ and real t > 0, let

[f]_{t} = {x ∈ ℝ^{n} : |f(x)| ≥ t},

denote the level sets of f. We always assume that our functions are such that the level sets [f]_{t} are compact for all t > 0.

The decreasing rearrangement, f, of f : ℝ^{n} → ℝ is defined by

f(x) = inf{t > 0 : V ([f]_{t}) < ω_{n}|x|^{n}},

where ω_{n}|x|^{n} is the n-dimensional volume of the ball of radius |x| in ℝ^{n}. The set [f]_{t} = {x ∈ ℝ^{n} : f(x) ≥ t} is a dilate of the unit ball, B = {x ∈ ℝ^{n} : |x| ≤ 1}, and its volume is equal to V ([f]_{t}); i.e.,

V ([f]_{t}) = V ([f]_{t}).

The functions f and f are equimeasurable, and therefore for all q ≥ 1

∥f∥_{q} = ∥f∥_{q}. (5.1)

Note that since f(x) depends only on the magnitude, |x|, of x (and not on the direction of x), there exists an increasing function f̂: (0, ∞) → ℝ defined by

f(x) = f̂(1/|x|). (5.2)

Observe that provided f is sufficiently smooth, f(x) = t implies (by definition of f) that V ([f]_{t}) = ω_{n}|x|^{n}, or equivalently, f̂(1/|x|) = t implies V ([f]_{t}) = ω_{n}|x|^{n}.

The following is needed in the proof of the main theorem:

sharp affine L_{p} sobolev inequalities 29

Lemma 5.1. If f ∈ H^{1,p}(ℝ^{n}) ∩ C^{∞}(ℝ^{n}), then

∫_{0}^{∞} V ([f]_{t})^{\frac{p(n−1)}{n}} (−V′([f]_{t}))^{1−p}dt = \frac{n^{1−p}}{ωn^{\frac{p−n}{n}}} ∫_{0}^{∞}(f̂′(s))^{p}s^{2p−n−1}ds, (5.3)

and ∥f∥_{q}^{q} = nω_{n} ∫_{0}^{∞} f̂(s)^{q}s^{−n−1}ds. (5.4)

Proof. Let t = f̂(s). Since f̂(1/|x|) = t implies V ([f]_{t}) = ω_{n}|x|^{n},

V ([f]_{t}) = s^{−n}ω_{n},

and hence

−V′([f]_{t}) = ns^{−n−1} \frac{ds}{dt}ω_{n}.

It follows that

V ([f]_{t})^{\frac{p(n−1)}{n}} (−V′([f]_{t}))^{1−p}dt = n^{1−p}s^{2p−n−1} (\frac{ds}{dt})^{−p} ds ω_{n}^{1− \frac{p}{n}} ,

which gives (5.3). To get (5.4) simply rewrite the defining integral for ∥f∥_{q}^{q} in polar coordinates. q.e.d.

6. Affine L_{p} Sobolev inequalities

Theorem 1′. Suppose p ∈ (0, 1), and q is given by \frac{1}{q} = \frac{1}{p} − \frac{1}{n}. If f ∈ H^{1,p}(ℝ^{n}), then (∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{−1/n} ≥ c_{2}∥f∥_{q}, (6.1) where the optimal c_{2} is given by

c_{2} = (\frac{n − p}{p − 1})^{1− \frac{1}{p}} (\frac{Γ(\frac{n}{p})Γ(n + 1 − \frac{n}{p})}{Γ(n + 1)})^{\frac{1}{n}} (\frac{nΓ(\frac{n}{2})Γ(\frac{p+1}{2})}{\sqrt{π}Γ(\frac{n+p}{2})})^{\frac{1}{p}}
= (2/nc_{4})^{\frac{1}{p}} (nω_{n})^{− \frac{1}{n}} c_{0}

and equality is attained when

f(x) = (a + |A(x − x_{0})|^{\frac{p}{p−1}} )^{1− \frac{n}{p}} ,

with A ∈ GL(n), real a > 0 and x_{0} ∈ ℝ^{n}.

Proof. It suffices to prove the inequality for compactly supported f ∈ C^{∞}(ℝ^{n}). For t > 0, consider the level sets of f,

[f]_{t} = {x ∈ ℝ^{n} : |f(x)| ≥ t}.

By Sard’s theorem, for almost all t > 0 the boundary, ∂[f]_{t}, of the level set is a C^{1} submanifold which has everywhere nonzero normal vector ∇f. Let dS_{t} denote the surface area element of ∂[f]_{t}. For t > 0, let K_{t} be the convex body constructed from f in Lemma 3.2.

We first need

∥D_{v}f∥_{p}^{p} = 2^{p}n ∫_{0}^{∞} V_{p}(K_{t}, v)dt. (6.2)

To see this simply note that by rewriting the integral, using (3.2), and then using (3.3b) we have

∥D_{v}f∥_{p}^{p} = ∫_{ℝ^{n}} |v · ∇f(x)|^{p}dx
= ∫_{0}^{∞} ∫_{∂[f]_{t}} |v · ∇f(x)|^{p}|∇f(x)|^{−1}dS_{t}(x) dt
= 2^{p}n ∫_{0}^{∞} V_{p}(f, t, v)dt
= 2^{p}n ∫_{0}^{∞} V_{p}(K_{t}, v)dt.

We need the fact that

V (K_{t})^{n−p} ≥ V ([f]_{t})^{(n−1)p}(−n^{−1}V′([f]_{t}))^{n(1−p)}. (6.3)

To see this, note that from (3.3a), definition (3.1), the Hölder inequality, definition (3.1) again, the extended Minkowski mixed volume inequality (1.2), and the co-area formula, we have

sharp affine L_{p} sobolev inequalities 31

V (K_{t})^{\frac{n−p}{np}} = V (K_{t})^{− \frac{1}{n}} V_{p}(f, t, K_{t})^{\frac{1}{p}}

= V (K_{t})^{− \frac{1}{n}} (\frac{1}{n} ∫_{∂[f]_{t}} h_{K_{t}}(ν(x))^{p}|∇f(x)|^{p−1}dS_{t}(x))^{\frac{1}{p}}

≥ n^{− \frac{1}{p}} V (K_{t})^{− \frac{1}{n}} (∫_{∂[f]_{t}} |∇f|^{−1}dS_{t})^{\frac{1−p}{p}} ∫_{∂[f]_{t}} h_{K_{t}}(ν(x))dS_{t}(x) = n^{1− \frac{1}{p}} V (K_{t})^{− \frac{1}{n}} (∫_{∂[f]_{t}} |∇f|^{−1}dS_{t})^{\frac{1−p}{p}} V_{1}([f]_{t}, K_{t}) ≥ n^{1− \frac{1}{p}} (∫_{∂[f]_{t}} |∇f|^{−1}dS_{t})^{\frac{1−p}{p}} V ([f]_{t})^{\frac{n−1}{n}} = n^{1− \frac{1}{p}} (−V′([f]_{t}))^{\frac{1−p}{p}} V ([f]_{t})^{\frac{n−1}{n}} . To complete the proof, observe that from (6.2), the Minkowski in-

equality for integrals, the affine inequality (2.2), (6.3), (5.3), (5.4), and (5.1),

(∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{− \frac{p}{n}}

= 2^{p}n (∫_{S^{n−1}} (∫_{0}^{∞} V_{p}(K_{t}, v) dt)^{− \frac{n}{p}} dv)^{− \frac{p}{n}} ≥ 2^{p}n ∫_{0}^{∞} (∫_{S^{n−1}} V_{p}(K_{t}, v)^{− \frac{n}{p}} dv)^{− \frac{p}{n}} dt

≥ \frac{2}{c_{4}n^{p/n}} ∫_{0}^{∞} V (K_{t})^{\frac{n−p}{n}} dt

≥ \frac{2n^{p−1− \frac{p}{n}}}{c_{4}} ∫_{0}^{∞} V ([f]_{t})^{\frac{(n−1)p}{n}} (−V′([f]_{t}))^{1−p}dt

= \frac{2n^{− \frac{p}{n}} ωn^{\frac{n−p}{n}}}{c_{4}} ∫_{0}^{∞}(f̂′(s))^{p}s^{2p−n−1}ds

≥ \frac{2}{nc_{4}} c(f̂)^{p}∥f∥_{q}^{p}

= \frac{2}{nc_{4}} c(f̂)^{p}∥f∥_{q}^{p},

where

c(f̂) = (∫_{0}^{∞}(f̂′(s))^{p}s^{2p−n−1}ds)^{\frac{1}{p}} (∫_{0}^{∞} f̂(s)^{q}s^{−n−1}ds)^{− \frac{1}{q}} .

Make the substitution t = 1/s and then define the function g by g(t) = f̂(1/t), to get

c(f̂) = (∫_{0}^{∞} |g′(t)|^{p}t^{n−1}dt)^{\frac{1}{p}} (∫_{0}^{∞} g(t)^{q}t^{n−1}dt)^{− \frac{1}{q}} ,

(recall that f̂ is increasing and thus g′ is always negative). Lemma 4.1 gives

c(f̂) ≥ c_{5}

and this proves the desired inequality. q.e.d.

Remark. The affine L_{p}-Sobolev inequality (0.4) implies the affine L_{p}-isoperimetric inequality (2.2). This can be seen by taking

f(x) = (1 + ρ_{K}(x)^{\frac{p}{1−p}} )^{1− \frac{n}{p}} ,

where ρ_{K}(x) = max{λ ≥ 0 : λx ∈ K} denotes the radial function of K.

A simple calculation shows that

∥D_{v}f∥_{p}^{p} = c_{6}V_{p}(K, v),

where

c_{6} = n2^{p} (\frac{n − p}{p − 1})^{p−1} Γ (\frac{n}{p}) Γ (n + 1 − \frac{n}{p}) /Γ(n).

Therefore, (2.2) is one of the consequences of the new inequality (0.4).

7. The L_{p} and affine L_{p} Sobolev inequalities

We will show that the new affine L_{p} Sobolev inequality is indeed stronger (and directly implies) the sharp L_{p} Sobolev inequality.

First observe that

(\frac{1}{nω_{n}} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{− \frac{p}{n}} ≤ \frac{2}{nc_{4}} ∫_{ℝ^{n}} |∇f(x)|^{p}dx. (7.1)

To see this, note that from the Hölder inequality and Fubini’s theorem we have

(\frac{1}{nω_{n}} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{− \frac{p}{n}}

≤ \frac{1}{nω_{n}} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{p}dv

= \frac{1}{nω_{n}} ∫_{S^{n−1}} ∫_{ℝ^{n}} |v · ∇f(x)|^{p}dxdv

= \frac{1}{nω_{n}} ∫_{ℝ^{n}} ∫_{S^{n−1}} |v · ∇f(x)|^{p}dvdx

= \frac{1}{nω_{n}} ∫_{S^{n−1}} |u_{0} · v|^{p}dv ∫_{ℝ^{n}} |∇f(x)|^{p}dx,

= \frac{2}{nc_{4}} ∫_{ℝ^{n}} |∇f(x)|^{p}dx,

where u_{0} is any fixed unit vector.

Now combine (7.1) with the affine L_{p} Sobolev inequality (6.1) to get:

(∫_{ℝ^{n}} |∇f|^{p} dx)^{\frac{1}{p}} ≥ [nc_{4}/2]^{1/p} (\frac{1}{nω_{n}} ∫_{S^{n−1}} ∥D_{v}f∥_{p}^{−n}dv)^{− \frac{1}{n}} ≥ c_{0} ∥f∥_{q}. (7.2)

From the equality conditions in the Hölder inequality it follows that equality in the left inequality in (7.2) occurs precisely when f is such that ∥D_{v}f∥_{p} is independent of v ∈ S^{n−1}.

8. An application to information theory

In this section we use Theorem 1 to prove a moment-entropy inequality for a Banach space–valued random variable X.

Let X be a random vector in ℝ^{n} with probability density g. Given λ > 0, the λ-Renyi entropy power N_{λ}(X) is defined by

N_{λ}(X) = \begin{cases}∥g∥λ^{\frac{λ}{1−λ}} & λ ≠ 1 \\ e^{− ∫ g log g} & λ = 1. \end{cases}

Observe that N_{1}(X) is the Shannon entropy power of X, and

lim_{λ→1} N_{λ}(X) = N_{1}(X).

A random vector X in ℝ^{n} with density function g is said to have finite r^{th}-moment, r > 0, if

∫_{ℝ^{n}} |x|^{r}g(x)dx < ∞.

If X is a random vector in ℝ^{n} with finite r^{th}-moment, and K is an originsymmetric convex body in ℝ^{n}, then the dual mixed volume Ṽ_{−r}(X, K) was defined in [32] by

Ṽ_{−r}(X, K) = \frac{n + r}{n} ∫_{ℝ^{n}} ∥x∥_{K}^{r}g(x) dx, (8.1)

where g is the density function of X and ∥ · ∥_{K} is the norm of the n-dimensional Banach space whose unit ball is K; i.e., for x ∈ ℝ^{n}

∥x∥_{K} = min{λ > 0 : x ∈ λK}.

The following is a special case of the dual Minkowski inequality established in [32].

Lemma 8.1. Suppose r > 0 and λ > \frac{n}{n+r}. If K is an originsymmetric convex body in ℝ^{n} and X is a random vector in ℝ^{n} with finite r^{th}-moment, then

Ṽ_{−r}(X, K)^{\frac{1}{r}} ≥ c_{7}[N_{λ}(X)/|K|]^{\frac{1}{n}} ,

where the best constant c_{7} is given by

c_{7} = \begin{cases}(1 − \frac{n(1−λ)}{rλ})^{\frac{1}{n(1−λ)}} (\frac{λ− \frac{n}{n+r}}{1−λ} )^{− \frac{1}{r}} (\frac{n}{r}B(\frac{n}{r}, \frac{1}{1−λ} − \frac{n}{r}))^{− \frac{1}{n}} & λ < 1, \\ (\frac{n+r}{re})^{\frac{1}{r}} Γ(\frac{n}{r} + 1)^{− \frac{1}{n}} & λ = 1, \\ (1 + \frac{n(λ−1)}{rλ})^{\frac{1}{n(1−λ)}} (\frac{λ− \frac{n}{n+r}}{λ−1} )^{− \frac{1}{r}} (\frac{n}{r}B(\frac{n}{r}, \frac{λ}{λ−1}))^{− \frac{1}{n}} & λ > 1. \end{cases}

Given a function f ∈ H^{1,p}(ℝ^{n}), let ∥ · ∥_{f,p} denote the associated Banach norm defined in the Introduction. If X is a random vector in the Banach space (ℝ^{n}, ∥ · ∥_{f,p}), the r^{th} moment of X is E(∥X∥_{f,p}^{r}). The following theorem gives a sharp lower bound of the moment E(∥X∥_{f,p}^{r}) in terms of the Renyi entropy power N_{λ}(X) and the L_{q} norm ∥f∥_{q}.

Theorem 8.2. Suppose 1 ≤ p < n, q = \frac{np}{n−p}, r > 0, and λ > \frac{n}{n+r}. If f ∈ H^{1,p}(ℝ^{n}) and X is a random vector in ℝ^{n} with finite r^{th}-moment, then

E(∥X∥_{f,p}^{r}) ≥ c_{8}N_{λ}(X)^{\frac{r}{n}} ∥f∥_{q}^{r},

where the best constant c_{8} = nc_{7}^{r}c_{1}^{−r}/(n + r).

Proof. Let B_{p}(f) denote the unit ball associated with the norm ∥ · ∥_{f,p}. Let g be the density function of X. Note that inequality (0.3) holds when p = 1 (see [42]), and thus inequality (0.2) holds when p = 1. From (8.1), Lemma 8.1, and (0.2), we have

E(∥X∥_{f,p}^{r}) = ∫_{ℝ^{n}} ∥x∥_{f,p}^{r} g(x) dx

= \frac{n}{n + r} Ṽ_{−r}(X, B_{p}(f))

≥ \frac{n}{n + r} c_{7}^{r}[N_{λ}(X)/|B_{p}(f)|]^{\frac{r}{n}}

≥ \frac{n}{n + r} c_{7}^{r}[N_{λ}(X)c_{1}^{−n}∥f∥_{q}^{n}]^{\frac{r}{n}}

= c_{8}[N_{λ}(X)∥f∥_{q}^{n}]^{\frac{r}{n}} .

q.e.d.

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Dept. of Math., Polytechnic University Brooklyn, New York 11201