On Evans’ and Choquet’s Theorems for Polar Sets

Abstract

By classical results of G.C. Evans and G. Choquet on “good” kernels G in potential theory, for every polar K_σ-set P, there exists a finite measure μ on P such that its potential Gμ is infinite on P, and a set P admits a finite measure μ on P such that Gμ is infinite exactly on P if and only if P is a polar G_δ-set. A known application of Evans’ theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. It is shown that, by an elementary “metric sweeping” of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = |x − y|^α−d on ${\mathbbm {R}^{d}}$, 2 < α < d, solves a long-standing open problem.

Appendix: : Application to the PWB-Method

In this section we shall recall the solution to the generalized Dirichlet problem for balayage spaces by the Perron-Wiener-Brelot method and how (a generalization of) Evans theorem enables us to use only harmonic upper and lower functions.

So let $(X,\mathcal {W})$ be a balayage space (where the assumption in Section ?? may be satisfied or not). Let ${\mathscr{B}}(X)$, $\mathcal {C}(X)$ denote the set of all Borel measurable numerical functions on X, continuous real functions on X, respectively. Let $\mathcal {P}$ be the set of all continuous real potentials for $(X,\mathcal {W})$, that is

$$ \mathcal{P}:=\{p\in \mathcal{W}\cap \mathcal{C}(X)\colon \exists\ q\in \mathcal{W}\cap \mathcal{C}(X), q>0, p/q\to 0 \text{ at infinity}\}, $$

see [2, 11] for a thorough treatment and, for example, [9, 13] for an introduction to balayage spaces.

We recall that, for all open sets V in X and x ∈ X, we have positive Radon measures $\varepsilon _{x}^{{V^c}}$ on X, supported by V^c and characterized by

$$ \int p d \varepsilon_{x}^{{V^c}}=R_{p}^{{V^c}}(x):=\inf\{w(x)\colon w\in \mathcal{W}, w\ge p\text{ on }V^{c}\} , \qquad p \in \mathcal{P}, $$

so that, obviously, $\varepsilon _{x}^{{V^c}}=\delta _{x}$ if x ∈ V^c. They lead to harmonic kernels H_V on X:

$$ H_{V}f(x):=\int f d\varepsilon_{x}^{{V^c}}, \qquad f\in\mathcal{B}^{+}(X), x\in X. $$

Let us now fix an open set U in X for which we shall consider the generalized Dirichlet problem (see [2, Chapter VII]). Let $\mathcal {V}(U)$ denote the set of all open sets V such that $\overline V$ is compact in U, and let ${~}^{\ast } {\mathscr{H}}(U)$ be the set of all functions $u\in {\mathscr{B}}(X)$ which are hyperharmonic on U, that is, are lower semicontinuous on U and satisfy

$$ -\infty< H_{V}u(x)\le u(x)\text{ for all }x\in V\in \mathcal{V}(U). $$

Then ${\mathscr{H}}(U):= {}^{\ast } {\mathscr{H}}(U)\cap (-{}^{\ast } {\mathscr{H}}(U))$ is the set of functions which are harmonic on U,

$$ \mathcal{H}(U) =\{h\in \mathcal{B}(X)\colon h|_{U}\in \mathcal{C}(U),\ H_{V}h(x)=h(x) \text{ for all }x\in V\in\mathcal{V}(U)\}. $$

A function $f\colon X\to \overline {\mathbbm {R}}$ is called lower $\mathcal {P}$-bounded, $\mathcal {P}$-bounded if there is some $p\in \mathcal {P}$ such that f ≥−p, |f|≤ p, respectively. For every numerical function f on X, we have the set of all upper functions

$$ {{\mathcal{U}}_{f}^{U}}:=\{u\in {}^\ast \mathcal{H}(U)\colon u\ge f\text{ on }{U^c},\ u\text{ lower } \mathcal{P} \text{-bounded and l.s.c.\ on } X\}, $$

the set ${{\mathscr{L}}_{f}^{U}}:=-{\mathcal {U}}_{-f}^{U}$ of all lower functions for f with respect to U, and the definitions

$$ {\overline H}_{f}^{U} :=\inf {\mathcal{U}}_{f}, \qquad {\underbar H}_{f}^{U}:=\sup {\mathcal{L}_{f}^{U}}. $$

For every $p\in \mathcal {P}$, there exists $q\in \mathcal {P}$, q > 0, such that p/q → 0 at infinity. Hence we may replace ${{\mathcal {U}}_{f}^{U}}$ by the smaller set of upper functions, which are positive outside a compact in X, without changing the infimum (if f ≥−p consider f + εq, ε > 0).

To avoid technicalities we state the resolutivity result (see [2, VIII.2.12]) only for $\mathcal {P}$-bounded functions:

Theorem A.1

For every $\mathcal {P}$-bounded $f\in {\mathscr{B}}(X)$,

$$ H_{U}f={\overline H}_{f}^{U}={\underbar H}_ f^{U} \in \mathcal{H}(U). $$

Remark A.2

Let us indicate how the general approach above yields the solution to the generalized Dirichlet problem for harmonic spaces in the way the reader may be more familiar with.

So let us assume for a moment that the harmonic measures $\varepsilon _{x}^{{V^c}}$, x ∈ V, for our balayage space are supported by ∂V so that (hyper)harmonicity on U does not depend on values on U^c, and let us identify functions on U with functions on X vanishing outside U.

Let f be a Borel measurable function on ∂U which is $\mathcal {P}$-bounded (amounting to boundedness if U is relatively compact) and let $\tilde {\mathcal {U}_{f}^{U}}$ be the set of all functions u on U which are hyperharmonic on U and satisfy

$$ \liminf\nolimits_{x\in U, x\to z}u(x)\ge f(z)\quad\text{ for every }z\in \partial U. $$

(A.1)

If $u\in {{\mathcal {U}}_{f}^{U}}$, then $\tilde u:=1_{U}u$ is hyperharmonic on U and $\liminf \nolimits _{x\to z}\tilde u(x)\ge u(z)\ge f(z)$ for every z ∈ ∂U, hence $\tilde u\in \tilde {{\mathcal {U}}_{f}^{U}}$. If, conversely, $\tilde u$ is a function in $\tilde {\mathcal {U}_{f}^{U}}$ then, extending it to X by $\liminf _{x\to z} u(z)$ for z ∈ ∂U and $\infty$ on $X\setminus \overline U$, we get a function $u\in {\mathcal {U}_{f}^{U}}$. Therefore Theorem A.1 yields that $h\colon x\mapsto \varepsilon _{x}^{U^{c}}(f)$, x ∈ U, is harmonic on U and

$$ h(x)=\inf \tilde{{\mathcal{U}}_{f}^{U}}(x)=\sup \tilde{\mathcal{L}_{f}^{U}}(x) \quad\text{ for every } x\in U. $$

Let ∂_regU denote the set of regular boundary points z of U, that is, z ∈ ∂U such that $\lim _{x\to z} H_{U}f(x)=f(z)$ for all $\mathcal {P}$-bounded $f\in \mathcal {C}(X)$, and let ∂_irrU be the set of irregular boundary points of U, ∂_irrU := ∂U ∖ ∂_regU.

Corollary A.3

Suppose that there is a lower semicontinuous function h₀ ≥ 0 on X which is harmonic on U and satisfies $h=\infty$ on ∂_irrU. Then

$$ H_{U}f=\inf {{\mathcal{U}}_{f}^{U}}\cap \mathcal{H}(U) =\sup {\mathcal{L}_{f}^{U}}\cap \mathcal{H}(U) \quad\text{ for every } \mathcal{P} \text{-bounded } f\in\mathcal{B}(X). $$

Proof

a) Let g be $\mathcal {P}$-bounded and lower semicontinuous on X. Then there exist $\mathcal {P}$-bounded φ_n in $\mathcal {C}(X)$, $n\in \mathbbm {N}$, such that φ_n ↑ g. For all z ∈ ∂_regU and $n\in \mathbbm {N}$,

$$ \liminf\nolimits_{x\to z} H_{U}g(x)\ge \liminf\nolimits_{x\to z} H_{U}\varphi_{n}(x)=\varphi_{n}(z), $$

and hence $\liminf _{x\to z} H_{U}g(x)\ge g(z)$. Clearly,

$$ h_{n}:=H_{U}g+(1/n) h_{0}\in \mathcal{H}(U) $$

satisfies $\lim _{x\to z} h_{n}(x)=\infty$ for all z ∈ ∂_irrU, and h_n is lower semicontinuous on X. Thus $h_{n}\in {\mathcal {U}_{f}^{U}}\cap {\mathscr{H}}(U)$.

b) Let $f\in {\mathscr{B}}(X)$ be $\mathcal {P}$-bounded, x ∈ X. There exists a decreasing sequence (g_n) of $\mathcal {P}$-bounded lower semicontinuous functions on X such that g_n ≥ f for every $n\in \mathbbm {N}$ and

$$ \int f d\varepsilon_{x}^{{U^c}}=\inf\nolimits_{n\in\mathbbm{N}} \int g_{n} d\varepsilon_{x}^{{U^c}}, $$

that is, $H_{U}f(x)=\inf _{n\in \mathbbm {N}} H_{U}g_{n}(x)$. Hence

$$ H_{U}f(x)=\inf\nolimits_{n\in\mathbbm{N}} (H_{U} g_{n} +(1/n)h_{0}) (x), $$

where $H_{U}g_{n}+(1/n)h_{0}\in {{\mathcal {U}}_{f}^{U}}\cap {\mathscr{H}}(U)$, by (a). Thus $H_{U}f=\inf {{\mathcal {U}}_{f}^{U}}\cap {\mathscr{H}}(U)$.

c) Further, $H_{U}f=-H_{U}(-f)=- \inf {\mathcal {U}}_{-f}^{U}\cap {\mathscr{H}}(U)= \sup {{\mathscr{L}}_{f}^{U}}\cap {\mathscr{H}}(U)$. □

Remarks A.4

1. For harmonic spaces, the result in Corollary A.3 has been proven in [6], where the solution to the generalized Dirichlet problem is obtained using controlled convergence (see also [17]).

2. In general, the set ∂_irrU is a semipolar F_σ-set. Of course, if $(X,\mathcal {W})$ satisfies Hunt’s hypothesis (H), that is, if every semipolar set is polar, then ∂_irrU is polar for every U. Let us note that (H) holds if X is an abelian group such that $\mathcal {W}$ is invariant under translations and $(X,\mathcal {W})$ admits a Green function having the local triangle property (see [14]).

By Theorem 1.1, we obtain that in this situation (which covers the classical case, many translation-invariant second order PDO’s as well as Riesz potentials, that is, α-stable processes, and many more general Lévy processes) the assumption of Corollary A.3 holds.