Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 27, 2002, 163–172

NONLINEAR PERTURBATION OF BALAYAGE SPACES

Azeddine Baalal and Wolfhard Hansen

Faculté des Sciences Aïn Chok, Département de Mathématiques Route El Jadida, B.P. 5366 Mâarif, Casablanca, Maroc Universität Bielefeld, Fakultät für Mathematik Universitätsstraße, D-33501 Bielefeld, Germany; hansen@mathematik.uni-bielefeld.de

Abstract. For a large class of nonlinear perturbations of balayage spaces existence and uniqueness of solutions to the Dirichlet problem are shown. In the special case of a harmonic space given by a linear second order partial differential operator L the perturbed equation is Lu − φ( · , u)μ = 0 where φ(x, t) is continuous in t ∈ R, the functions φ_{c} := sup{|φ( · , t)| : −c ≤ t ≤ c}, c > 0, are locally μ-Kato, i.e., yield continuous real L-potentials ^{U}G_{L}^{φ_{c}μ}, and the functions t ↦ φ(x, t), x ∈ X, have a weak form of joint lower Lipschitz property, i.e., ψ := sup_{s<t}(φ( · , t) − φ( · , s))^{−}/(t − s) is locally μ-Kato and perturbation by −ψμ still leads to a 𝒫 -harmonic structure.

1. Introduction and basic notions

Recently the Dirichlet problem for nonlinear perturbation of partial differential equations of the type

Lu − uφ( · , u)μ = 0

(L being a linear elliptic or parabolic operator of second order) has been studied in the potential-theoretic setting of a harmonic space ([BM], [BBM]).

We shall be able to weaken the assumptions, our model case being

Lu − φ( · , u)μ = 0

(cf. (2.2)), to get better results, and, nevertheless, have shorter proofs.

In fact, our method works even for balayage spaces (see [BH]) covering in addition nonlocal situations as e.g. given by Riesz potentials. So we shall study nonlinear perturbation of balayage spaces. The reader who is mainly interested in the PDE case leading to harmonic spaces (which can be viewed as balayage spaces where harmonic measures for open sets live on the boundaries) might consult [He], [CC], [BH], [K], [HH] and [Bo].

It will be convenient to assume that our balayage space has a base of regular sets. We adopt the same notations as in [Ha] and recall briefly the basic definitions:

2000 Mathematics Subject Classification: Primary 31C45, 35J60, 35K55, 45K05.

Let X be a locally compact space with countable base. For every open set U in X, let ℬ(U) denote the set of all numerical Borel measurable functions on U. Further, 𝒞 (U) will denote the space of all continuous real functions on U and 𝒦 (U) the set of all functions in 𝒞 (U) having compact support in U. Occasionally, functions on U will be identified with functions on X which are zero on U^{c}. Finally, given any set 𝒜 of functions let 𝒜_{b} (𝒜^{+} resp.) denote the set of all functions in 𝒜 which are bounded (positive resp.).

Let 𝒰 be a base of relatively compact open subsets of X and, for every U ∈ 𝒰 , let H_{U} be a kernel on X such that H_{U}(x, ·) = ε_{x} for every x ∈ U^{c} and H_{U}1_{U} = 0. Let us suppose that 𝒰 is stable with respect to finite intersections (by [BH, Remark VII.3.2.4] this is no restriction of generality). Define

(1.1)  𝒲 := {v | v : X → [0, ∞] l.s.c., H_{U}v ≤ v for every U ∈ 𝒰 }

and, for every numerical function f ≥ 0 on X, let

R_{f} := inf{v ∈ 𝒲 : v ≥ f}.

A function s ∈ 𝒞^{+}(X) is called strongly (𝒲 )-superharmonic if, for every U ∈ 𝒰 , H_{U}s < s on U.

Then (H_{U})_{U∈𝒰} is a family of (regular) harmonic kernels and (X, 𝒲 ) is a balayage space provided the following holds (where U, V ∈ 𝒰 ): (H_{1}) Given x ∈ X, lim_{U↓{x}} H_{U}f(x) = f(x) for all f ∈ 𝒦 (X) or R_{1_{{x}}} is l.s.c.

at x. (H′_{2}) H_{V} H_{U} = H_{U} if V ⊂ U. (H_{3}) For every f ∈ ℬ_{b}(X) with compact support, the function H_{U}f is continuous

on U. (H′_{4}) For every f ∈ 𝒦 (X), the function H_{U}f is continuous on \overline{U}. (H′_{5}) There exists a strongly superharmonic function s ∈ 𝒞^{+}(X).

In the following (X, 𝒲 ) will always denote a balayage space associated with a family (H_{U})_{U∈𝒰} of regular harmonic kernels. For simplicity let us suppose that there exists a strictly positive bounded function in 𝒲 . This is no real loss of generality, since we may always replace 𝒲 by {w/s : w ∈ 𝒲 }, s being an arbitrary strictly positive function in 𝒲 ∩ 𝒞 (X). (Note, however, that we would have to replace ℬ_{b}(X) by the space of all s-bounded functions.)

For every open subset U of X let ℋ (U) denote the set of all harmonic functions on U, i.e.,

ℋ (U) := {h ∈ ℬ(X) : h|_{U} continuous, H_{V} h(x) = h(x)
for every x ∈ V ∈ 𝒰 , \overline{V} ⊂ U\}.

One way of defining the convex cone 𝒫(X) of all continuous real potentials is the following:

𝒫(X) := {p ∈ 𝒲 ∩ 𝒞 (X) : 0 ≤ g ≤ p, g ∈ ℋ ^{+}(X) ⟹ g = 0}.

We recall from [BH] that for every open subset U of X, regular or not, bounded or not, there is a harmonic kernel H_{U}. It is characterized by

H_{U}p = inf{q ∈ 𝒫(X) : q ≥ p on U^{c}} (p ∈ 𝒫(X)).

By definition, a sequence (x_{n}) in U converging to a point z ∈ ∂U is called regular if lim_{n→∞} H_{U}f(x_{n}) = f(z) for every f ∈ 𝒦 (X). If it is regular, then lim_{n→∞} H_{U}f(x_{n}) = f(z) even for every f ∈ ℬ_{b}(X) such that the restriction of f on U^{c} is continuous at z. The set U is regular, if every sequence in U converging to a boundary point of U is regular. In particular, every U ∈ 𝒰 is regular by (H′_{4}).

Let us fix a potential kernel K_{X} for (X, 𝒲 ), i.e., K_{X} is a kernel such that

(1.2) K_{X}f ∈ 𝒫(X) ∩ ℋ (X \ supp(f)) for f ∈ ℬb^{+}(X) with compact support.

For every open subset U of X, we define a kernel K_{U} by

(1.3)  K_{U} := K_{X} − H_{U}K_{X}.

Since obviously K_{U}(x, ·) = 0 for every x ∈ U^{c}, we may view K_{U} as being a kernel on X or a potential kernel on U (restricting H_{V} (x, ·) on U for x ∈ V ∈ 𝒰 with \overline{V} ⊂ U we obtain a family of harmonic kernels on U). We recall that K_{U} is a compact operator on ℬ_{b}(X) if U is relatively compact (this follows easily from [Ha, Lemma 10.1] and (1.3)). Moreover,

(1.4)  K_{U} = K_{V} + H_{V} K_{U}

for all open U, V with V ⊂ U.

A function f ∈ ℬ(X) is called a Kato function (with respect to K_{X}) if K_{X}M_{f} is a potential kernel (M_{f} denotes multiplication by f) or, equivalently, if

K_{X}(1_{U_{n}}f^{±}) ∈ 𝒞 (X)

for a sequence (U_{n}) of open sets covering X. Of course, every locally bounded function in ℬ(X) is a Kato function. More generally, every f ∈ ℬ(X) which is locally bounded by a Kato function is a Kato function.

Remark 1.1. If the balayage space (X, 𝒲 ) is given by a second order differential operator L with Green function G_{L} (such that LG_{L}( · , y) = −δ_{y}), then potential kernels are associated with (Kato) measures μ by K_{X}f = G_{L}^{fμ} := ∫ G_{L}( · , y)f(y) μ(dy).

2. Main result

Let φ be a Borel measurable real function on X × R such that the functions t ↦ φ(x, t), x ∈ X, are continuous on R. For every x ∈ X and real c > 0, we define

\begin{aligned}φ_{c}(x) & := max\{|φ(x, t)| : −c ≤ t ≤ c\}, \\ ψ(x) & := sup\{(\frac{φ(x, t) − φ(x, s)}{t − s})^{−} : −∞ < s < t < ∞\}. \end{aligned}

We note that ψ: X → [0, ∞] is the smallest function such that the functions t ↦ φ(x, t) + ψ(x)t, x ∈ X, are increasing. Moreover,

ψ(x) = sup_{t∈R}(\frac{∂φ}{∂t}(x, t))^{−}

if ∂φ/∂t exists.

Let us fix a relatively compact open subset U of X and assume the following:

(i) For every c > 0, K_{X}(1_{U}φ_{c}) ∈ 𝒞 (X), i.e., 1_{U}φ_{c} is a Kato function. (ii) K_{X}(1_{U}ψ) ∈ 𝒞 (X), i.e., 1_{U}ψ is a Kato function. (iii) Perturbation of (X, 𝒲 ) by −1_{U}ψ (with respect to K_{X}) yields a balayage

space (X, 𝒲̃ ) (see [Ha]).

Remarks 2.1. 1. If (i) and (ii) hold for U, then (i) and (ii) hold for any open V ⊂ U.

2. Of course (i) holds if φ is locally bounded. Further, (ii) holds if (∂φ/∂t)(x, t) exists and the functions ((∂φ/∂t)(x, ·))^{−}, x ∈ U, are uniformly bounded.

3. Assuming that 1_{U}ψ is a Kato function, (iii) holds provided there exist s ∈ 𝒲 and u ∈ ℬ^{+}(X) such that

v := s + K_{X}u ∈ 𝒞 (X), ψv ≤ u on U

and, for every V ∈ 𝒰 , {H_{V} s < s}∪{K_{V} (1_{U}ψv) < K_{V} u} = V [Ha, Theorem 6.4]). A special case would be p := K_{X}1 ∈ C(X) strongly superharmonic and ψ < 1/p on U (s := 0, u := 1).

4. If (X, 𝒲 ) is parabolic, (iii) is already a consequence of (ii). Indeed, suppose that 1_{U}ψ is a Kato function. By [Ha, Lemma 10.1], K_{X}M_{1_{U} ψ} is a compact operator on ℬ_{b}(X) and therefore, by [Ha, Theorem 10.2, Lemma 10.3],

L := ∑_{m=0}^{∞}(K_{X}M_{1_{U} ψ})^{m}

is a bounded operator on ℬ_{b}(X). Choose a strongly superharmonic bounded s ∈ 𝒲 ∩ 𝒞 (X) and define v := Ls, u := 1_{U}ψv. Then v = s + K_{X}u and ψv = u on U. Thus (iii) holds by the preceding remark.

We define a (nonlinear) operator K_{U}^{φ}: ℬ_{b}(X) → ℬ_{b}(X) by

K_{U}^{φ}v := K_{U}(φ( · , v)) (v ∈ ℬ_{b}(X)).

Of course, K_{U}^{φ} lives on ℬ_{b}(U): K_{U}^{φ}v depends only on the restriction of v on U and vanishes on U^{c}.

Given f ∈ ℬ_{b}(X), we shall say that a function H_{U}^{φ}f ∈ ℬ_{b}(X) is a (generalized) solution to the perturbed Dirichlet problem associated with U and f provided

(2.1)  H_{U}^{φ}f + K_{U}^{φ}H_{U}^{φ}f = H_{U}f.

In the situation of Remark 1.1 equation (2.1) implies that

(2.2)  LH_{U}^{φ}f − φ( · , H_{U}^{φ}f)μ = 0.

Moreover, H_{U}^{φ}f has essentially the same boundary behavior as H_{U}f. If e.g. U is regular, then K_{U}^{φ}H_{U}^{φ}f tends to zero at ∂U whence lim_{x→z} H_{U}^{φ}f(x) = f(z) for any z ∈ ∂U where f is continuous.

Our main result is the following:

Theorem 2.2. 1. For every f ∈ ℬ_{b}(X) there exist a unique generalized solution H_{U}^{φ}f to the perturbed Dirichlet problem. Moreover,

\begin{aligned}−(I − K_{U}M_{ψ})^{−1} & (H_{U}f^{−} + K_{U}(φ( · , 0)^{+})) ≤ H_{U}^{φ}f \\ & ≤ (I − K_{U}M_{ψ})^{−1}(H_{U}f^{+} + K_{U}(φ( · , 0)^{−})), \end{aligned}

H_{V}^{φ}H_{U}^{φ}f = H_{U}^{φ}f for every open V with \overline{V} ⊂ U, H_{U}^{φ}f is continuous on U, and lim_{n→∞} H_{U}^{φ}f(x_{n}) = f(z) for every regular sequence (x_{n}) in U converging to a point z ∈ ∂U where the restriction of f to the complement of U is continuous.

2. If f, g ∈ ℬ_{b}(X) such that f ≤ g, then H_{U}^{φ}f ≤ H_{U}^{φ}g.

3. If a bounded sequence (f_{n}) in ℬ_{b}(X) converges pointwise to a function f, then lim_{n→∞} H_{U}^{φ}f_{n} = H_{U}^{φ}f.

Obviously, K_{V}^{φ} for V ⊂ U is not changed if we replace φ by 1_{(U∩{ψ<∞})×R}φ (note that K_{X}(1_{{ψ=∞}}) = 0 by (ii)). Therefore we may assume without loss of generality that φ(x, ·) = 0 for every x ∈ U^{c} and that ψ is a real function. To prove Theorem 2.2 we may in addition suppose that |φ| ≤ 1 and all functions t ↦ φ(x, t), x ∈ X, are increasing. Indeed, fix c > 0 and define

φ̃(x, t) := \frac{φ(x, t_{c}) + t_{c}ψ(x)}{φ_{c}(x) + cψ(x) + 1} (x ∈ X, t ∈ R)

where t_{c} := min(max(−c, t), c).

Obviously, |φ̃| ≤ 1, φ̃ = 0 on U^{c} ×R and every function t ↦ φ̃(x, t), x ∈ X, is increasing and continuous. By (iii), for every relatively compact open W in X, the operator I − K_{W} M_{ψ} is invertible, (I − K_{W} M_{ψ})^{−1} = ∑_{m=0}^{∞}(K_{W} M_{ψ})^{m}, and

(2.3)  H̃_{W} = (I − K_{W} M_{ψ})^{−1}H_{W}

is the harmonic kernel for W with respect to (X, 𝒲̃ ). By (i) and (ii),

K′_{X} := K_{X}M_{φ_{c}+cψ+1}

is a potential kernel with respect to (X, 𝒲 ). Clearly, K_{X}M_{ψ} = K′_{X}M_{ψ/(φ_{c}+cψ+1)} . This implies that the balayage space (X, 𝒲̃ ) is obtained perturbing (X, 𝒲 ) by −ψ/(φ_{c} + cψ + 1) with respect to K′_{X}. We finally note that there exists a unique potential kernel K̃_{X} for (X, 𝒲̃ ) such that, for every relatively compact open W in X,

K̃_{W} = (I − K_{W} M_{ψ})^{−1}K′_{W}

(this follows from (2.3) and [Ha, Proposition 10.5]).

Fix a > 0 and let f ∈ ℬ_{b}(X) such that |f| ≤ a. Since the functions H_{U}1 and K_{U}|φ( · , 0)| are bounded, we may choose b > 0 such that

(I − K_{U}M_{ψ})^{−1}(aH_{U}1 + K_{U}|φ( · , 0)|) ≤ b.

Then

(I − K_{U}M_{ψ})^{−1}(H_{U}|f| + K_{U}|φ( · , 0)|) ≤ b.

Moreover, fix g ∈ ℬ_{b}(X) and c ≥ b such that |g| ≤ c.

Suppose now that the statements of Theorem 2.2 hold for (X, 𝒲̃ ), K̃_{X}, and φ̃ (defined using this constant c). Then

|H̃φ̃ ; Uf| ≤ H̃_{U}|f| + K̃_{U}|φ̃( · , 0)| = (I − K_{U}M_{ψ})^{−1}(H_{U}|f| + K_{U}|φ( · , 0)|) ≤ b ≤ c.

Further,

K′_{U}(φ̃( · , g)) = K_{U}(φ( · , g) + ψg) = K_{U}(φ( · , g)) + K_{U}M_{ψ}g

whence

g +K_{U}(φ( · , g)) = (I −K_{U}M_{ψ})g +K′_{U}(φ̃( · , g)) = (I −K_{U}M_{ψ})(g +K̃_{U}(φ̃( · , g))).

Thus

g + K_{U}^{φ}g = H_{U}f ⟺ g + K̃φ̃ ; Ug = H̃_{U}f ⟺ g = H̃φ̃ ; Uf.

This implies that H_{U}^{φ}f = H̃φ̃ ; Uf and that Theorem 2.2 holds for (X, 𝒲 ), K_{X}, and φ.

3. Existence and uniqueness of the solution to the perturbed Dirichlet problem

As indicated in the previous section we shall assume from now on that |φ| ≤ 1 and all functions t ↦ φ(x, t), x ∈ X, are continuous and increasing. For every real function v on X we define a function Φ(v) by

Φ(v)(x) := φ(x, v(x)) (x ∈ X).

Clearly, for every v ∈ ℬ_{b}(X),

K_{U}^{φ}v = K_{U}(Φ(v)).

Our assumption on φ implies that |Φ(v)| ≤ 1 for every v ∈ ℬ_{b}(X). Moreover, Φ(v) ≤ Φ(w) on {v ≤ w}, and (Φ(v_{n})) converges pointwise to Φ(v) if (v_{n}) converges pointwise to v.

Lemma 3.1. Let (v_{n}) be a sequence in ℬ_{b}(X). Then there exists a subsequence (w_{n}) of (v_{n}) such that the sequence (K_{U}^{φ}w_{n}) in ℬ_{b}(X) converges uniformly. Moreover, if (v_{n}) converges pointwise to a function v, then (K_{U}^{φ}v_{n}) converges uniformly to K_{U}^{φ}v.

Proof. (Φ(v_{n})) is a bounded sequence in ℬ_{b}(X) and K_{U} is a compact operator on ℬ_{b}(X). This implies the first statement. The second statement now follows from the continuity of the functions t ↦ φ(x, t) and the fact that K_{U} is a kernel.

Proposition 3.2. The operator I + K_{U}^{φ}: ℬ_{b}(X) → ℬ_{b}(X) is surjective.

Proof. We fix g ∈ ℬ_{b}(X) and consider the mapping T from ℬ_{b}(X) into ℬ_{b}(X) defined by

T u := g − K_{U}^{φ}u.

By Lemma 3.1, T is continuous and T (ℬ_{b}(X)) is relatively compact. By Schauder’s fixed point theorem there exists u ∈ ℬ_{b}(X) such that T u = u, i.e., we have u + K_{U}^{φ}u = g. Thus I + K_{U}^{φ} is surjective.

Let ^{∗}ℋ ^{+}(U) denote the set of all functions s ∈ ℬ^{+}(X) such that s is l.s.c. on U and H_{V} s ≤ s for every V ∈ 𝒰 with \overline{V} ⊂ U. If s ∈ ^{∗}ℋ ^{+}(U), then obviously 1_{U}s ∈ ^{∗}ℋ ^{+}(U).

Lemma 3.3. Let v, w, g ∈ ℬ_{b}(X) and s ∈ ^{∗}ℋ ^{+}(U) such that v + K_{U}w = g ≤ s and {w > 0} ⊂ {v ≥ 0}. Then v ≥ g − s. In particular, v ≥ 0 if

g ∈ ^{∗}ℋb^{+}(U).

Proof. Obviously, K_{U}w ≤ g ≤ s on {v ≥ 0}, hence on {w > 0}. Consequently K_{U}w ≤ s and v = g − K_{U}w ≥ g − s.

Proposition 3.4. Let v, w, g ∈ ℬ_{b}(X) and s ∈ ^{∗}ℋ ^{+}(U) such that |g| ≤ s, vw ≥ 0, and v + K_{U}w = g. Then |v − g| ≤ s.

Proof. Apply Lemma 3.3 to v, w, g and −v, −w, −g.

Corollary 3.5. The operator I + K_{U}^{φ}: ℬ_{b}(X) → ℬ_{b}(X) is bijective.

Proof. By Proposition 3.2, the operator I + K_{U}^{φ} is surjective. To show that it is injective, we fix v_{1}, v_{2} in ℬ_{b}(U) such that v_{1} + K_{U}^{φ}v_{1} = v_{2} + K_{U}^{φ}v_{2} . Taking v := v_{1} − v_{2} and w := Φ(v_{1}) − Φ(v_{2}) we have vw ≥ 0 and v + K_{U}w = 0, hence |v| ≤ 0 by Proposition 3.4 (taking g = s = 0). Thus v_{1} = v_{2} .

An immediate consequence is the following:

Theorem 3.6. For every f ∈ ℬ_{b}(X), there exists a unique solution H_{U}^{φ}f to the perturbed Dirichlet problem.

4. Properties of the solution to the perturbed Dirichlet problem

As before we suppose that U is a relatively compact open subset of X, |φ| ≤ 1, and the functions t ↦ φ(x, t), x ∈ X, are continuous and increasing.

Proposition 4.1. Let f ∈ ℬ_{b}(X). Then H_{V}^{φ}H_{U}^{φ}f = H_{U}^{φ}f for every V ∈ 𝒰 with \overline{V} ⊂ U. Moreover, lim_{n→∞} H_{U}^{φ}f(x_{n}) = f(z) for every regular sequence (x_{n}) in U converging to a point z ∈ ∂U where the restriction of f to the complement of U is continuous.

Proof. Define h := H_{U}^{φ}f. Then h is continuous on U, since K_{U}^{φ}H_{U}f ∈ 𝒞 (U) and H_{U}f is harmonic on U. Moreover, for every V ∈ 𝒰 with \overline{V} ⊂ U,

h + K_{V}^{φ}h = h + K_{U}^{φ}h − H_{V} K_{U}^{φ}h = H_{U}f − H_{V} (H_{U}f − h) = H_{V} h.

Fix z ∈ ∂U such that f|_{U^{c}} is continuous at z and let (x_{n}) be a regular sequence in U such that lim_{n→∞} x_{n} = z. Then lim_{n→∞} H_{U}f(x_{n}) = f(z) and we conclude from (1.3) that lim_{n→∞} K_{U}g(x_{n}) = 0 for every g ∈ ℬ_{b}(X). Thus lim_{n→∞} H_{U}^{φ}f(x_{n}) = f(z) by (2.1).

Moreover, we easily obtain the following (note that a combination of (1) and (2) yields the first inequalities in Theorem 2.2):

Proposition 4.2. Let f, f_{1}, f_{2}, … ∈ ℬ_{b}(X). Then the following holds:

1. −K_{U}(φ( · , 0)^{+}) ≤ H_{U}^{φ}0 ≤ K_{U}(φ( · , 0)^{−}).

2. −H_{U}((f_{1} − f_{2})^{−}) ≤ H_{U}^{φ}f_{1} − H_{U}^{φ}f_{2} ≤ H_{U}((f_{1} − f_{2})^{+}).

3. If f_{2} ≤ f_{1} then H_{U}^{φ}f_{2} ≤ H_{U}^{φ}f_{1} .

4. If the sequence (f_{n}) is bounded and converges pointwise to a function f, then lim_{n→∞} H_{U}^{φ}f_{n} = H_{U}^{φ}f.

Proof. 1. Let v = H_{U}^{φ}0 and s_{±} = K_{U}(φ( · , 0)^{±})). Then v ∈ ℬ_{b}(X), s_{±} ∈ ^{∗}ℋb^{+}(U), and taking w := Φ(v) − Φ(0) we obtain that

v + K_{U}w = −K_{U}(φ( · , 0)) = s_{−} − s_{+} =: g.

Of course, g ≤ s_{−} and −g ≤ s_{+} . Since {w > 0} = {Φ(v) > Φ(0)} ⊂ {v ≥ 0}, Lemma 3.3 implies that v ≥ g − s_{−} = −s_{+} . Since {−w > 0} ⊂ {−v ≥ 0}, we obtain that −v ≥ −g − s_{+} = −s_{−}. Thus −s_{+} ≤ v ≤ s_{−}.

2. Let v_{1} = H_{U}^{φ}f_{1} , v_{2} = H_{U}^{φ}f_{2} , v = v_{1} − v_{2} , and w = Φ(v_{1}) − Φ(v_{2}). Then {w > 0} ⊂ {v ≥ 0} and v + K_{U}w = H_{U}(f_{1} − f_{2}) =: g. In particular,

g ≤ H_{U}((f_{1} − f_{2})^{+}) ∈ ^{∗}ℋb^{+}(U) and −g ≤ H_{U}((f_{1} − f_{2})^{−}) ∈ ^{∗}ℋb^{+}(U). Thus, by

Lemma 3.3,

\begin{aligned}v & ≥ g − H_{U}((f_{1} − f_{2})^{+}) = −H_{U}((f_{1} − f_{2})^{−}), \\ −v & ≥ −g − H_{U}((f_{1} − f_{2})^{−}) = −H_{U}((f_{1} − f_{2})^{+}). \end{aligned}

3. Immediate consequence of (2), since (f_{1} − f_{2})^{−} = 0 if f_{2} ≤ f_{1} .

4. For every n ∈ N,

(4.1)  H_{U}^{φ}f_{n} + K_{U}^{φ}H_{U}^{φ}f_{n} = H_{U}f_{n}.

Of course, lim_{n→∞} H_{U}f_{n} = H_{U}f and the sequence (H_{U}^{φ}f_{n}) is bounded by (2). Moreover, by Lemma 3.1, there exists a subsequence (g_{n}) of (f_{n}) such that the sequence (K_{U}^{φ}H_{U}^{φ}g_{n}) is convergent. So we conclude from (4.1) that the sequence (H_{U}^{φ}g_{n}) converges to a function G ∈ ℬ_{b}(X). Letting n tend to infinity we obtain from (4.1) that

G + K_{U}^{φ}G = H_{U}f.

Thus H_{U}^{φ}f = G = lim_{n→∞} H_{U}^{φ}g_{n}. By a general argument on subsequences, this shows that in fact lim_{n→∞} H_{U}^{φ}f_{n} = H_{U}^{φ}f.

Proposition 4.3. Let h ∈ ℬ_{b}(X) such that, for every z ∈ ∂U, lim_{y→z,y∉U} h(y) = h(z) and lim_{n→∞} h(x_{n}) = h(z) for every regular sequence (x_{n}) converging to z. Moreover, suppose that U is covered by subsets V ∈ 𝒰 satisfying H_{V}^{φ}h = h. Then h = H_{U}^{φ}h.

Proof. Define g := h + K_{U}^{φ}h and let V ∈ 𝒰 be a subset of U such that H_{V}^{φ} = h. Then h + K_{V}^{φ}h = H_{V} h is harmonic on V and K_{U}^{φ}h − K_{V}^{φ}h is harmonic on U ∩ V . Therefore g is harmonic on U ∩ V and we conclude that g is harmonic on U. So g − H_{U}h is a function in ℬ_{b}(X) which is harmonic on U, equal to zero on U^{c}, and tends to zero along every regular sequence converging to a boundary point of U. This implies that g − H_{U}h = 0 whence h = H_{U}^{φ}h.

172 Azeddine Baalal and Wolfhard Hansen

Corollary 4.4. If the restriction of f ∈ ℬ_{b}(X) to the complement of U is continuous at every z ∈ ∂U, then H_{U}^{φ}f is the only function g ∈ ℬ_{b}(X) such that g = f on U^{c}, H_{V}^{φ}g = g for every V ∈ 𝒰 with \overline{V} ⊂ U, and lim_{n→∞} g(x_{n}) = f(z) for every regular sequence (x_{n}) convergent to a point z ∈ ∂U.

Finally, let us suppose that our assumptions (i)–(iii) hold for every relatively compact open subset U in X. Then, for every open subset W of X, we may define ^{φ}ℋ (W) to be the set of all h ∈ ℬ_{b}(X) such that h is continuous on W and H_{V}^{φ}h = h for every V ∈ 𝒰 with \overline{V} ⊂ W. The following sheaf property is an immediate consequence of Proposition 4.3.

Corollary 4.5. The set {^{φ}ℋ (W) : W open ⊂ X} is a sheaf, i.e., for every family (W_{i})_{i∈I} of open subsets in X,

φℋ ( ⋃_{i∈I} W_{i}) = ⋂_{i∈I} ^{φ}ℋ (W_{i}).

References

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Received 16 November 2000