Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 18, 1993, 65–75

QUASIADDITIVITY OF CAPACITY AND MINIMAL THINNESS

Hiroaki Aikawa

Kumamoto University, Faculty of Science, Department of Mathematics Kumamoto 860, Japan

Abstract. It is well known that a capacity C is countably subadditive, i.e.

C(E) ≤ ∑ C(E_{j}), E = ∪E_{j}.

We shall prove that the reverse inequality, up to a multiplicative constant, holds for some decomposition of E, provided there is a measure comparable to C. Such a property will be referred to as quasiadditivity.

As an application, we shall show that the Green energy for a uniformly Δ-regular domain is quasiadditive with respect to the Whitney decomposition of the domain. The Hardy inequality due to Ancona [3, (1)] will be a main tool. We shall apply the quasiadditivity of the Green energy to obtain a refined Wiener criterion for minimal thinness in an NTA domain.

1. Introduction

Let X be a locally compact Hausdorff space and let k be a nonnegative lower semicontinuous function on X × X. We refer to k as a kernel and define the capacity C_{k} by

C_{k}(E) = inf {∥μ∥ : k(·, μ) ≥ 1 on E},

where k(·, μ) = ∫_{X} k(·, y) dμ(y). It is well known that C_{k} is countably subadditive, i.e.

C_{k}(E) ≤ ∑ C_{k}(E_{j}), E = ⋃ E_{j}.

In this article we shall show that the reverse inequality, up to a multiplicative constant, holds for some decomposition of E.

Definition. Let {Q_{j}} and {Q_{j}^{∗}} be families of Borel subsets of X such that:

(i) Q_{j} ⊂ Q_{j}^{∗}, (ii) X = ∪Q_{j}, (iii) Q_{j}^{∗} do not overlap so often, i.e., ∑ χ_{Q_{j}^{∗}} ≤ N.

Then we say that {Q_{j}, Q_{j}^{∗}} is a quasidisjoint decomposition of X. We sometimes suppress Q_{j}^{∗} and simply write {Q_{j}}.

We write M for a positive constant whose value may change from one occurrence to the next. If M^{−1}f ≤ g ≤ Mf for two positive quantities f and g, then we write f ≈ g. This constant is referred to as the constant of comparison.

1991 Mathematics Subject Classification: Primary 31B15; Secondary 31B25.

Definition. Let σ be a (Borel) measure on X. We say that σ is comparable to C_{k} with respect to {Q_{j}} if

(1)  σ(Q_{j}) ≈ C_{k}(Q_{j}) for every Q_{j},
(2)  σ(E) ≤ MC_{k}(E) for every Borel set E.

Definition. We say that the kernel k has the Harnack property with respect to {Q_{j}, Q_{j}^{∗}} if

k(x, y) ≈ k(x′, y)

for x, x′ ∈ Q_{j} and y ∈ X \ Q_{j}^{∗} with the constant of comparison independent of Q_{j}.

Our main theorem is as follows.

Theorem 1. Let {Q_{j}, Q_{j}^{∗}} be a quasidisjoint decomposition of X. Suppose the kernel k has the Harnack property with respect to {Q_{j}, Q_{j}^{∗}}. If there is a measure σ comparable to C_{k} with respect to {Q_{j}, Q_{j}^{∗}}, then for every E ⊂ X

(3)  C_{k}(E) ≈ ∑ C_{k}(E ∩ Q_{j}).

We shall say that C_{k} is quasiadditive with respect to {Q_{j}, Q_{j}^{∗}} if (3) holds. The proof of the theorem will be carried out in Section 2 in the same spirit as in [2].

It is, in general, difficult to find a measure comparable to a given capacity. In [2] we found such a measure for the Riesz capacity with the aid of a certain weighted norm inequality. In this paper we shall employ a different device— Hardy’s inequality—to obtain a measure comparable to a generalized Green energy. The most general Hardy inequality is given for a uniformly Δ-regular domain by Ancona [3]. By B(x, r) we denote the open ball with radius r and center at x. We shall say that D ⊂ R^{d} is uniformly Δ-regular if there are constants r_{0} > 0 and ε_{1} , 0 < ε_{1} < 1, such that, for all x ∈ ∂D and all 0 < r < r_{0} ,

w_{x,r} ≤ 1 − ε_{1} on B(x, r/2) ∩ D,

where w_{x,r} is the harmonic measure of ∂B(x, r) ∩ D in the region B(x, r) ∩ D ([3, Definition 2]). The following lemma is a consequence of [3, Theorem 1 and Proposition 1].

Lemma A (Hardy’s inequality). Assume that D is uniformly Δ-regular. Then, there is a positive constant M depending only on D such that

∫_{D} |\frac{ψ(x)}{δ(x)}|^{2}dx ≤ M ∫_{D} |∇ψ(x)|^{2}dx for all ψ ∈ W_{0}^{1,2}(D),

where δ(x) = dist(x, ∂D) and W_{0}^{1,2}(D) stands for the usual Sobolev space, namely the completion of C_{0}^{∞}(D) with norm (∫_{D}(|ψ|^{2} + |∇ψ|^{2}) dx)^{1/2} .

We let G(x, y) be the Green function normalized by

ΔG(·, y) = −δ_{y},

where the left hand side denotes the distributional Laplacian of G(·, y) and δ_{y} the point mass at y. Let u be a nonnegative superharmonic function on D. For a compact subset K of D we let R̂_{u}^{K} be the regularized reduced function of u with respect to K. Observe that R̂_{u}^{K} is a Green potential, G(·, λ_{u}^{K}). The energy

γ_{u}(K) = ∫∫ G(x, y) dλ_{u}^{K}(x) dλ_{u}^{K}(y)

is called the Green energy of K relative to u. For an open subset V we let

γ_{u}(V) = sup {γ_{u}(K) : K is compact, K ⊂ V },

and then for a general subset E

γ_{u}(E) = inf {γ_{u}(V) : V is open, E ⊂ V }.

The quantity γ_{u}(E) is also called the Green energy relative to u. If u ≡ 1, then γ_{u} is the usual Green capacity C_{G} (see [15, p. 174–177]). If D = {x = (x_{1}, …, x_{d}) : x_{1} > 0} and u(x) = x_{1} , then γ_{u} is the Green energy defined by Essén and Jackson [11, Definition 2.2]. Let k(x, y) = G(x, y)/(u(x)u(y)). Then it is not so difficult to see that γ_{u}(E) = C_{k}(E) (see [12]). Thus Theorem 1 applies to γ_{u}. Let δ(x) = dist(x, ∂D). We define the measure σ_{u} on D by

σ_{u}(E) = ∫_{E} (\frac{u(x)}{δ(x)})^{2}dx.

Theorem 2. Let D be a uniformly Δ-regular domain and let {Q_{j}} be the Whitney decomposition of D. Suppose a positive superharmonic function u satisfies

(4)  sup_{Q_{j}} u ≤ M_{0} inf_{Q_{j}} u

with M_{0} independent of Q_{j}. Then σ_{u} is comparable to γ_{u} with respect to {Q_{j}} and γ_{u} is quasiadditive with respect to {Q_{j}}, i.e. γ_{u}(E) ≈ ∑ γ_{u}(E ∩ Q_{j}).

For each Whitney cube Q_{j} we let x_{j} be the center of Q_{j}, r_{j} the diameter of Q_{j} and t_{j} = dist(Q_{j}, ∂D). We put u_{j} = u(x_{j}). By cap we denote the Newton capacity if d ≥ 3; the logarithmic capacity if d = 2. If E ⊂ Q_{j}, then γ_{u}(E) ≈ u_{j}^{2} cap(E) for d ≥ 3, and γ_{u}(E) ≈ u_{j}^{2}/log (4r_{j}/ cap(E)) for d = 2 (see Lemma 1 below). Hence Theorem 2 shows that the Green energy γ_{u} is estimated by the summation of the ordinary capacities of E ∩ Q_{j}.

Let h be a positive harmonic function. Then the Harnack principle shows that u(x) = min {h(x)^{a}, b} with 0 < a ≤ 1 and b > 0 satisfies (4). Hence we have the following corollaries.

Corollary 1. Let D and {Q_{j}} be as in Theorem 2. Then the Green capacity C_{G} = γ_{1} is quasiadditive with respect to {Q_{j}}, i.e.

C_{G}(E) ≈ ∑ C_{G}(E ∩ Q_{j}) ≈ \{ \begin{matrix}∑ cap(E ∩ Q_{j}) & if d ≥ 3, \\ ∑ \frac{1}{log (4r_{j}/ cap(E ∩ Q_{j}))} & if d = 2. \end{matrix}

Corollary 2. Let D and {Q_{j}} be as in Theorem 2. Let x_{0} ∈ D and let g(x) = min {G(x, x_{0}), 1}. Then γ_{g} is quasiadditive with respect to {Q_{j}}, i.e. with g_{j} = g(x_{j})

γ_{g}(E) ≈ ∑ γ_{g}(E ∩ Q_{j}) ≈ \{ \begin{matrix}∑ g_{j}^{2} cap(E ∩ Q_{j}) & if d ≥ 3, \\ ∑ \frac{g_{j}^{2}}{log (4r_{j}/ cap(E ∩ Q_{j}))} & if d = 2. \end{matrix}

In [14] Jerison and Kenig introduced the notion of NTA domains. A bounded domain D is called NTA when there exist positive constants M and r_{1} such that (a) Corkscrew condition. For any z ∈ ∂D, r < r_{1} there exists a point A_{r}(z) ∈ D

such that M^{−1}r < |A_{r}(z) − z| < r and δ(A_{r}(z)) > M^{−1}r.

(b) The complement of D satisfies the corkscrew condition.

(c) Harnack chain condition. If ε > 0 and x_{1} and x_{2} belong to to D, δ(x_{j}) > ε and |x_{1} − x_{2}| < Cε, then there exists a Harnack chain from x_{1} and x_{2} whose length depends on C, but not ε.

In view of (b) we see that an NTA domain D is uniformly Δ-regular. It is known that the Martin boundary of D is homeomorphic to the Euclidean boundary ∂D and every boundary point is minimal ([14]). We shall apply Corollary 2 to a characterization of minimally thin sets in an NTA domain. For details see Section 4.

Acknowledgement. I would like to thank Professor Stephen Gardiner for bringing Ancona’s paper [4] to my attention. I would like to thank Professor Alano Ancona for the comments on Section 4.

2. Proof of Theorem 1

Proof of Theorem 1. Let E ⊂ X. We may assume that C_{k}(E) < ∞. By definition we can find a measure μ such that k(·, μ) ≥ 1 on E and ∥μ∥ ≤ 2C_{k}(E). For each Q_{j} we let

μ_{j} = μ|_{Q_{j}^{∗}}, μ′_{j} = μ|_{X\Q_{j}^{∗}}.

We have the following two cases:

(a) k(x, μ_{j}) ≥ \frac{1}{2} for all x ∈ E ∩ Q_{j}.

(b) k(x, μ′_{j}) ≥ \frac{1}{2} for some x ∈ E ∩ Q_{j}.

If (a) holds, then C_{k}(E ∩ Q_{j}) ≤ 2∥μ_{j}∥ by definition. Since Q_{j}^{∗} do not overlap so often, we obtain

(5)  ∑′ C_{k}(E ∩ Q_{j}) ≤ 2 ∑′ ∥μ_{j}∥ ≤ M∥μ∥ ≤ MC_{k}(E),

where ∑′ denotes the summation over all Q_{j} for which (a) holds. If (b) holds, then the Harnack property of k yields that k(·, μ) ≥ k(·, μ′_{j}) ≥ M on Q_{j}, so that

k(·, μ) ≥ M on ∪′′ Q_{j},

where ∪′′ denotes the union over all Q_{j} for which (b) holds. Hence

C_{k}(∪′′Q_{j}) ≤ M∥μ∥.

Since σ is comparable to C_{k}, it follows from (1), (2) and the countable additivity of σ that

\begin{aligned}∑′′ C_{k}(E ∩ Q_{j}) & ≤ ∑′′ C_{k}(Q_{j}) ≤ M ∑′′ σ(Q_{j}) \\ & ≤ Mσ(∪′′Q_{j}) ≤ MC_{k}(∪′′Q_{j}) ≤ M∥μ∥ ≤ MC_{k}(E). \end{aligned}

This, together with (5), completes the proof.

Corollary 3. Let k, {Q_{j}, Q_{j}^{∗}} and σ be as in Theorem 1. If E is a union of Q_{j}, then C_{k}(E) ≈ σ(E).

Remark. It is not so difficult to give an L^{p} capacity version of Theorem 1. For 1 < p < ∞ and a fixed measure ν on X we let

C_{k,p}(E) = inf { ∫_{X} f^{p}dν : k(·, fν) ≥ 1 on E, f ≥ 0}.

Then the same argument yields that if k has the Harnack property with respect to {Q_{j}, Q_{j}^{∗}} and there is a measure σ comparable to C_{p} with respect to {Q_{j}, Q_{j}^{∗}}, then

C_{k,p}(E) ≈ ∑ C_{k,p}(E ∩ Q_{j}).

See [2, Theorem 1].

3. Proof of Theorem 2

In this section we let D be a uniformly Δ-regular domain. Let {Q_{j}} be the Whitney decomposition of D and let Q_{j}^{∗} be the double of Q_{j} (see [20, Chapter VI]). Then {Q_{j}, Q_{j}^{∗}} is a quasidisjoint decomposition of D. Throughout this section we suppose that a positive superharmonic function u satisfies (4).

We shall show that σ_{u} is comparable to γ_{u} with respect to {Q_{j}}. We can easily verify the first condition (1) by recalling the identity γ_{u}(E) = C_{k}(E), where k(x, y) = G(x, y)/(u(x)u(y)).

Lemma 1. Let Q_{j} be a Whitney cube and let r_{j} and u_{j} be as in the introduction. If E ⊂ Q_{j}, then

γ_{u}(E) ≈ \{ \begin{matrix}u_{j}^{2} cap(E) & if d ≥ 3, \\ \frac{u_{j}^{2}}{log (4r_{j}/ cap(E))} & if d = 2. \end{matrix}

In particular, γ_{u}(Q_{j}) ≈ σ_{u}(Q_{j}) ≈ u_{j}^{2}r_{j}^{d−2} .

Proof. From (4), we deduce the estimate

k(x, y) ≈ \{ \begin{matrix}u_{j}^{−2}|x − y|^{2−d} & if d ≥ 3, \\ u_{j}^{−2} log \frac{4r_{j}}{|x − y|} & if d = 2, \end{matrix}

for x, y ∈ Q_{j}, which proves the lemma.

Let K be a compact subset of D. Then (4) yields that u is bounded on K. Hence R̂_{u}^{K} is a bounded Green potential of a measure supported on K. By the next lemma we can apply Lemma A to R̂_{u}^{K}.

Lemma 2. Let K be a compact subset of D. Suppose that μ is a measure on K and that v = G(·, μ) is a bounded Green potential. Then v ∈ W_{0}^{1,2}(D) and

∫∫ G(x, y) dμ(x) dμ(y) = ∫_{D} |∇v|^{2}dx.

In fact, Lemma 2 can be extended to more general situations (see e.g. [7, Satz 7.2]); but the above form is sufficient for our purpose. Let us now prove (2).

Lemma 3. Let E be a Borel subset of D. Then

σ_{u}(E) ≤ Mγ_{u}(E).

Proof. Let K be a compact subset of E and write v_{K} = R̂_{u}^{K} = G(·, λ_{u}^{K}). Then

γ_{u}(K) = ∫∫ G(x, y) dλ_{u}^{K}(x) dλ_{u}^{K}(y) = ∫_{D} |∇v_{K}|^{2}dx

by Lemma 2. Since v_{K} = u q.e. on K and hence a.e. on K, it follows from Lemmas A and 2 that

γ_{u}(E) ≥ γ_{u}(K) = ∫_{D} |∇v_{K}|^{2}dx ≥ M ∫_{D} (\frac{v_{K}}{δ} )^{2}dx ≥ M ∫_{K} (\frac{u}{δ})^{2}dx = Mσ_{u}(K).

Since K ⊂ E is an arbitrary compact set, we have the required inequality.

Proof of Theorem 2. We have seen in Lemmas 1 and 3 that the measure σ_{u} is comparable to γ_{u} with respect to {Q_{j}}. Suppose y ∈ D \ Q_{j}^{∗}. Then G(·, y) is a positive harmonic function in the interior of Q_{j}^{∗}. Hence the Harnack principle yields that, for x, x′ ∈ Q_{j}, G(x, y) ≈ G(x′, y), so that k(x, y) ≈ k(x′, y) by (4). Thus the kernel k has the Harnack property with respect to {Q_{j}, Q_{j}^{∗}}. Therefore, the quasiadditivity of the Green energy γ_{u} follows from Theorem 1.

Remark. If D is a Liapunov domain, then g(x) ≈ δ(x) and Corollary 2 follows from the weak L^{1} estimate of Naïm’s Θ kernel (cf. [2], [18], [19] and [21]). For a general uniformly Δ-regular domain, however, the Θ kernel is not necessarily weak (1, 1); it may not be even a standard kernel.

4. Minimally thin sets in an NTA domain

In this section we let D be an NTA domain in R^{d}. Let D_{1} = {x ∈ D : G(x, x_{0}) > 1}. By definition g(x) = G(x, x_{0}) for x ∈ D \ D_{1} . Since the Martin boundary of D is homeomorphic to the Euclidean boundary ∂D and every boundary point is minimal, it follows that the ratio

K(x, y) = \frac{G(x, y)}{g(y)}

becomes continuous in the extended sense on D × \overline{D}. By the same symbol we denote the continuous extension on D × \overline{D}. Sometimes we write K_{y} for K(·, y). By definition K_{y}(x_{0}) = 1 for y ∈ \overline{D} \ D_{1} . If y ∈ ∂D, then K_{y} is a minimal harmonic function on D. Let E ⊂ D and let y ∈ ∂D. We say that E is minimally thin at y if the regularized reduced function R̂_{K_{y}}^{E} is a Green potential, or equivalently there is a finite measure μ on \overline{D} such that μ(\{y\}) = 0 and

(6)  K_{y} ≤ K(·, μ)  on E.

By the definition of Green energy we have

(7)  R̂_{g}^{E}(x_{0}) = γ_{g}(E)

for any set E. This, together with Corollary 2, yields a refined Wiener criterion for minimal thinness in terms of usual capacity. For each Whitney cube Q_{j} we let we put R_{j}(y) = |x_{j} − y| and A_{j}(y) = A_{R_{j}(y)}(y), where we recall x_{j} is the center of Q_{j} and that A_{r}(y) is the point appearing in the corkscrew condition for y ∈ ∂D.

Theorem 3. Suppose y ∈ ∂D and E ⊂ D. Then E is minimally thin at y if and only if

∑ (\frac{g_{j}}{g(A_{j}(y))})^{2}R_{j}(y)^{2−d} cap(E ∩ Q_{j}) < ∞ if d ≥ 3,

∑ (\frac{g_{j}}{g(A_{j}(y))})^{2} \frac{1}{log (4r_{j}/ cap(E ∩ Q_{j}))} < ∞ if d = 2.

Corollary 4. Let y ∈ ∂D. If a measurable set E is minimally thin at y, then

(8)  ∫_{E} (\frac{g(x)}{g(A_{|x−y|}(y))})^{2} \frac{|x − y|^{2−d}}{δ(x)^{2}} dx < ∞.

Remark. For a half space Essén [9] has first introduced the refined Wiener criterion. (See also [2].) He used the weak L^{1} estimate due to Sjögren [19]. As remarked before, the weak L^{1} estimate needs not hold for an NTA domain.

Definition. Suppose y ∈ ∂D and E ⊂ D. We say that E is minimally thin at y for harmonic functions if there is a finite measure μ concentrated on ∂D such that μ({y}) = 0 and (6) holds.

Theorem 4. Suppose y ∈ ∂D and E ⊂ D. Then E is minimally thin at y for harmonic functions if and only if

∑_{E∩Q_{j}≠∅} (\frac{g_{j}}{g(A_{j}(y))})^{2}(\frac{r_{j}}{R_{j}(y)})^{d−2} < ∞.

Corollary 5. Let y ∈ ∂D and let 0 < ϱ < 1. If E is minimally thin at y for harmonic functions, then E_{ϱ} = ∪_{x∈E}B(x, ϱδ(x)) satisfies (8) with E_{ϱ} replacing E.

Remark. A set E is said to determine the point measure at y ∈ ∂D if, for every finite measure μ concentrated on ∂D, (6) implies that μ({y}) > 0 ([6], [8], [17] and [19]). We note that E is minimally thin at y for harmonic functions if and only if E does not determine the point measure at y. Obviously if E is minimally thin at y for harmonic functions, then it is minimally thin; but the converse is not necessarily true.

Let us remark that Hayman [13, p. 481 and Theorem 7.37] defined sets “rarefied for harmonic functions”. We note that the term “rarefied sets” was introduced by Essén and Jackson [11]. A set is “rarefied” in the terminology of Lelong-Ferrand [16] if and only if it is “semirarefied” in the terminology of Essén and Jackson. A set rarefied for harmonic functions corresponds to a rarefied set defined by Essén and Jackson. See [10] for further information.

Remark. A sequence {z_{i}} ⊂ D is said to be separated if |z_{i} − z_{i′}| ≥ Mδ(z_{i}) for i ≠ i′ with some positive constant M. It is easy to see that {z_{i}} is separated if and only if the number of points z_{i} included in a Whitney cube Q_{j} is bounded by a positive constant independent of Q_{j}. Thus Theorem 4 implies Ancona’s criterion for a set to determine the point measure [4, Theorem 7.4]. Ancona first indicated that the Hardy inequality can be used. His short description of the proof was stated only for Lipschitz domains, but it seems to work for NTA domains. He also gave further discussions for hyperbolic graphs [4] and Denjoy domains [5].

Remark. If D is a Liapunov domain, then g(x) ≈ δ(x) ([21]), so that Theorems 3, 4 and their Corollaries are generalization of the results of Beurling [6], Maz’ya [17], Dahlberg [8], Sjögren [19] and Essén [10, Section 2].

5. Proofs of Theorems 3 and 4

In this section let D be an NTA domain. Let Θ(x, y) = G(x, y)/(g(x)g(y)). It is known that Θ(x, y) has a continuous extension on \overline{D} × \overline{D}. By the same symbol we denote the continuous extension. The kernel Θ is referred to as the Naïm’s Θ kernel for D ([18]). By definition Θ is symmetric.

Lemma 4. For 0 < R < r_{1} and y ∈ ∂D we let θ_{y}(R) = Θ(A_{R}(y), y) with A_{R}(y) appearing in the corkscrew condition. Then

(9)  θ_{y}(R) ≈ R^{2−d}g(A_{R}(y))^{−2}.

If x ∈ D and |x − y| = R, then Θ(x, y) ≈ θ_{y}(R). Moreover, if r ≈ R, then θ_{y}(r) ≈ θ_{y}(R).

Proof. This is essentially proved in [1, Lemma 3.3]; but for the sake of convenience we give a proof. We shall use the same notation as in [14]; Δ(y, r) stands for the surface ball B(y, r) ∩ ∂D and ω^{x}(·) for the harmonic measure of D evaluated at x. If x = x_{0} , then we write simply ω(·) for ω^{x_{0}}(·).

It is well known that

K(x, y) = lim_{r→0} \frac{ω^{x}(Δ(y, r))}{ω(Δ(y, r))}

(cf. [14, Theorem 5.5]). If we carefully consider the above convergence, then we have, for 0 < r ≤ R,

K(A_{R}(y), y) ≈ \frac{ω^{A_{R}(y)}(Δ(y, r))}{ω(Δ(y, r))}.

On the other hand, if r ≈ R, then ω^{A_{R}(y)}(Δ(y, r)) ≈ 1. Therefore, letting \frac{1}{2}R ≤ r ≤ R, we obtain from [14, Lemma 4.8] and the Harnack principle that

K(A_{R}(y), y) ≈ \frac{1}{ω(Δ(y, r))} ≈ \frac{1}{r^{d−2}g(A_{r}(y))} ≈ R^{2−d}g(A_{R}(y))^{−1}.

By definition (9) follows. Moreover the boundary Harnack principle (cf. [14, Lemma 4.10]) says that if x ∈ D and |x − y| = R, then

\frac{K(x, y)}{K(A_{R}(y), y)} ≈ \frac{g(x)}{g(A_{R}(y))},

which implies that Θ(x, y) ≈ Θ(A_{R}(y), y) = θ_{y}(R). The last assertion follows from the Harnack principle. The lemma is proved.

Proof of Theorem 3. In the same way as in [1] we can show that E is minimally thin at y if and only if

∑_{i} R̂_{K_{y}}^{E_{i}}(x_{0}) < ∞,

where E_{i} = {x ∈ E : 2^{−i} ≤ |x − y| < 2^{1−i}}. By Lemma 4 we see that K_{y} ≈ θ_{y}(2^{−i})g on {x ∈ D : 2^{−i} ≤ |x − y| < 2^{1−i}}. Hence it follows from (7) that the above condition is equivalent to

∑_{i} θ_{y}(2^{−i})γ_{g}(E_{i}) < ∞.

Invoking Corollary 2, we can rewrite the above condition as

(10)  ∑_{j} θ_{y}(R_{j}(y))γ_{g}(E ∩ Q_{j}) < ∞,

where we recall R_{j}(y) = |x_{j} − y| and x_{j} is the center of Q_{j}. Hence Lemmas 1 and 4 assert that (10) leads to the condition of Theorem 3. Thus the theorem is proved.

Proof of Corollary 4. In view of Lemmas 3 and 4, we see that (10) implies that

∑ R_{j}(y)^{2−d}g(A_{R_{j}(y)}(y))^{−2} \frac{g_{j}^{2}}{t_{j}^{2}} ∫_{E∩Q_{j}} dx < ∞,

which is equivalent to (8).

Proof of Theorem 4. It is not so difficult to see that E is minimally thin at y for harmonic functions if and only if Ẽ = ∪_{E∩Q_{j}≠∅}Q_{j} is minimally thin at y ([10]). Hence Lemma 1 and Theorem 3 readily imply the theorem.

References

[1] Aikawa, H.: On the minimal thinness in a Lipschitz domain. - Analysis 5, 1985, 347–382.

[2] Aikawa, H.: Quasiadditivity of Riesz capacity. - Math. Scand. 69, 1991, 15–30.

Quasiadditivity of capacity and minimal thinness 75

[3] Ancona, A.: On strong barriers and an inequality of Hardy for domains in R^{n}. - J. London Math. Soc. (2) 34, 1986, 274–290.

[4] Ancona, A.: Positive harmonic functions and hyperbolicity. - In Potential Theory— Surveys and Problems. Lecture Notes in Mathematics 1344. Springer-Verlag, 1987, 1–23.

[5] Ancona, A.: Sur la frontière de Martin des domaines de Denjoy. - Ann. Acad. Sci. Fenn. Ser. A I Math. 15, 1990, 259–271.

[6] Beurling, A.: A minimum principle for positive harmonic functions. - Ann. Acad. Sci. Fenn. Ser. A I Math. 372, 1965.

[7] Constantinescu, C., and A. Cornea: Ideale Ränder Riemannscher Flächen. - Springer-Verlag, 1963.

[8] Dahlberg, B.E.J.: A minimum principle for positive harmonic functions. - Proc. London Math. Soc. (3) 33, 1976, 238–250.

[9] Essén, M.: On Wiener conditions for minimally thin sets and rarefied sets. - In Complex Analysis, Articles dedicated to A. Pfluger on the occasion of his 80th birthday. Birkhäuser, 1988, 41–50.

[10] Essén, M.: On minimal thinness, reduced functions and Green potentials. - Preprint, 1990.

[11] Essén, M., and H.L. Jackson: On the covering properties of certain exceptional sets in a half-space. - Hiroshima Math. J. 10, 1980, 233–262.

[12] Fuglede, B.: Le théorème du minimax et la théorie fine du potentiel. - Ann. Inst. Fourier (Grenoble) 15, 1965, 65–87.

[13] Hayman, W.K., and P.B. Kennedy: Subharmonic functions, Volume 1. - Academic Press, 1976; Volume 2, Academic Press, 1989.

[14] Jerison, D.S., and C.E. Kenig: Boundary behavior of harmonic functions in nontangentially accessible domains. - Adv. in Math. 46, 1982, 80–147.

[15] Landkof, N.S.: Foundations of modern potential theory. - Springer-Verlag, 1972.

[16] Lelong-Ferrand, J.: Étude au voisinage de la frontière des fonctions surharmoniques positive dans un demi-espace. - Ann. Sci. École Norm. Sup. 66, 1949, 125–159.

[17] Maz’ya, V.G.: Beurling’s theorem on a minimum principle for positive harmonic functions. - Zapiski Nauchnykh Seminarov LOMI 30, 1972, 76–90 (Russian); English translation: J. Soviet Math. 4, 1975, 367–379.

[18] Naïm, L.: Sur le rôle de la frontière de R.S. Martin dans la théorie du potentiel. - Ann. Inst. Fourier (Grenoble) 7, 1957, 183–281.

[19] Sjögren, P.: Weak L_{1} characterization of Poisson integrals, Green potentials and H^{p} spaces. - Trans. Amer. Math. Soc. 233, 1977, 179–196.

[20] Stein, E.M.: Singular integrals and differentiability properties of functions. - Princeton University Press, 1970.

[21] Widman, K.-O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. - Math. Scand. 21, 1967, 17–37.

Received 9 December 1991