ON BOUNDARY REGULARITY OF THE DIRICHLET PROBLEM FOR PLANE DOMAINS

TOSHIYUKI SUGAWA

Abstract. In this paper, we propose a notion of the local harmonic measure decay (LHMD) property with exponent α at finite boundary points of open sets Ω in the Riemann sphere ℂ̂. Using this property, we show that Green’s function of Ω is Hölder continuous with exponent α at such a point as well as the boundary regularity of the Dirichlet problem in Ω for the usual Laplacian at the point in the sense of Hölder continuity with exponent less than α. We further explain that the LHMD property can be regarded as a localization of the notion of uniform perfectness for the boundary. We also provide several applications to the theory of conformal mappings.

1. Introduction

1.1. Hölder regularity of a boundary point. We consider the boundary regularity of the bounded solution of the Dirichlet problem in a plane domain (or, more generally, an open set) Ω in the sense of Perron-Wiener-Brelot:

(1.1) Δu = 0 in Ω and u = φ on ∂Ω,

where Δ denotes the usual Laplacian (∂/∂x)^{2}+(∂/∂y)^{2} and φ is a given bounded function. Throughout this article, we consider only bounded harmonic functions for solutions of the Dirichlet problem.

For a positive constant α, a point a ∈ ∂Ω \ {∞} will be called an α-Hölder regular boundary point of Ω if the solution u of the Dirichlet problem in Ω is Hölder continuous with exponent α at a whenever the boundary function φ is Hölder continuous with exponent α. More precisely, this means that if φ(ζ) = φ(a) + O(|ζ − a|^{α}) as ζ → a in ∂Ω then u(z) = φ(a) + O(|z − a|^{α}) as z → a in Ω.

The Hölder regularity of boundary points has been investigated by many authors even for more general uniformly elliptic (possibly non-homogeneous) linear partial differential equations (see [1], [14] and their bibliographies).

When Ω is a disk or a half plane (a ball or a half space in the higher dimensional case), it is known that each finite boundary point is α-Hölder regular for any α with 0 < α < 1 (see [34, Proposition 3.4] for the unit disk in the plane, or [1, Theorem 3.1] for more general differential equations). On the one hand, any boundary point of the disk is never 1-Hölder regular (see §6.9 and Remark made there).

When the boundary is of class C^{2} and of bounded curvature, O. D. Kellogg [24] showed the same result (in the case of 3-dimensional space).

Date: March 8, 2001.

1991 Mathematics Subject Classification. Primary 31A15; Secondary 30C85.

Key words and phrases. Green’s function, uniformly perfect, boundary regularity, Hölder continuity, Dirichlet problem.

For domains with exterior cone condition, K. Miller [31] gave a satisfactory result on the Hölder regularity of the boundary for uniformly elliptic partial differential equations. We remark that C. Pucci [35] obtained partial results in this direction earlier.

In the case when the boundary is not necessarily smooth, however, such a property seems to be less known. As one exception, we should draw the attention of the reader to the paper [19] by A. Hinkkanen, in which he proved α-Hölder regularity of each point of a boundary continuum for 0 < α < 1/2. Actually, he obtained more detailed, almost sharp global results on the Dirichlet solutions for bounded plane domains each boundary component of which has at least a definite size of diameter.

We should note also that, in the case of higher dimensions (n > 2), there is pioneer work of V. G. Maz’ya (see [29], [30] and also [16]) in this direction even for uniformly elliptic differential operators.

1.2. Control of boundary behaviour. In order to control the boundary behaviour of a harmonic function, the notion of barrier has been effectively used. In this article, we propose “local harmonic measures” as good substitutions for barriers. The author learned this notion from the paper [3] of A. Ancona. The precise definition will be given in Section 2. The local harmonic measures have advantage in the sense that they enjoy

1. monotonicity with respect to domain extension and circular symmetrization, and 2. availability to use classical conformal mappings.

In fact, by virtue of these properties, we deduce many estimates for local harmonic measures in Section 5. These have applications to the theory of conformal mappings, see Section 7.

We investigate the boundary regularity using the local harmonic measures. As a result, we give a condition, called “LHMD property,” for a boundary point to be Hölder regular in terms of the local harmonic measures in Section 2.

1.3. LHMD property and uniform perfectness. The global LHMD property, which means the LHMD property in a uniform sense, is shown to be same as the uniform Δregularity in the sense of Ancona [3] (see Theorem 3.1 below). Ancona [3] proved that a closed set E in ℝ^{n} has uniformly Δ-regular complement if and only if E satisfies a uniform capacity density condition with respect to the Newtonian capacity in the case n ≥ 3. The Newtonian capacity being replaced by the logarithmic capacity, the same result holds even if n = 2 as noted by several authors (cf. [12]). The proof for the case n = 2 requires slightly different (but standard) techniques from that of [3], so we include it in Section 3 for the convenience of the reader as well as for getting explicit estimates with the emphasis on the relation with the LHMD property.

Noting a result of Pommerenke (see §4.1), we see that the global LHMD property of a domain is nothing but the uniform perfectness of the boundary. Here, a closed set E in the Riemann sphere ℂ̂ containing at least two points is said to be uniformly perfect if there exists a positive constant c such that E ∩ {z ∈ ℂ; cr < |z − a| < r} ≠ ∅ for any point a ∈ E and 0 < r < diam E, where diam E denotes the Euclidean diameter of E (diam E = +∞ if ∞ ∈ E).

This notion was introduced by [6], and systematically investigated by Ch. Pommerenke [32], [33]. Nowadays, a number of equivalent conditions for the uniform perfectness are known. Uniformly perfect objects naturally arise in many branches of complex analysis.

Indeed, the limit set of a non-elementary Kleinian group of Lehner type (containing finitely generated case, see [40]), the Julia set of a rational function of degree at least two (see [8]), and some kind of self-similar fractals (see [39] and [38]) are uniformly perfect. For a survey on the uniform perfectness, see [41].

As a consequence of the above chain of ideas, we give another proof for the known fact that Green’s function of a domain with uniformly perfect boundary is Hölder continuous up to the boundary (see Corollary 6.3).

1.4. Structure of this article. Now we briefly describe the organization of the present paper.

Section 2 gives the definition and fundamental properties of local harmonic measures as well as basic notions in potential theory. We prove there our key result (Theorem 2.2) saying that the LHMD property with exponent α at a finite boundary point a implies Hölder regularity with exponent less than α at the point a.

Section 3 is devoted to (analytic) characterizations of the LHMD property such as a (lower) capacity density condition with concrete estimates for related quantities.

In Section 4, we further show that the global LHMD property of Ω is actually equivalent to the uniform perfectness of the boundary ∂Ω (Theorem 4.1). In connection with the uniform perfectness, we also give local, geometric conditions which are sufficient or necessary for the LHMD property at a given point.

Section 5 treats several geometric conditions which ensure the LHMD property with a given exponent at a boundary point such as the exterior circle condition, the generalized exterior wedge condition. We also consider the case when the domain is bounded by (positively oriented) sufficiently smooth Jordan curves whose curvature is bounded below. Then, we require a global property (Theorem 5.4) of simple plane curves with curvature bounded below. For the convenience of the reader, we shall give a proof for that property in the appendix (Section 9). This might be of independent interest.

Section 6 explains how to globalize the local results obtained in preceding sections in the case when Ω satisfies the uniform LHMD property with exponent α. Specifically, we show α-Hölder continuity of Green’s function of the domain and the boundedness of the harmonic extension operator H^{Ω} on the Lipschitz spaces of exponent less than α.

As an application, we have a lower estimate of Hausdorff dimension of a totally disconnected uniformly perfect set (Corollary 6.4). Compare with the similar result in [41].

In Section 7, we will give several applications of Hölder continuity of Green’s function to the theory of conformal mappings of finitely connected bounded domains. As results, we present a natural generalization of a theorem due to M. Masumoto [28], and L^{p}- integrability conditions for the derivatives of conformal mappings from the domain onto a standard domain. The latter is closely related to the Brennan conjecture when the domain is simply connected.

In Section 8, we will give a simple example of a bounded domain carrying Green’s function which is Hölder (actually Lipschitz) continuous up to the boundary, whereas the boundary is not uniformly perfect. This example answers a question raised by Siciak in [37] and produces another example of domain which preserves the global Markov inequality but not the local Markov inequality (see [26]).

Section 9 is additional and serves a proof of the above-mentioned result on simple plane curves with curvature bounded below.

1.5. Acknowledgements. I sincerely thank Hiroaki Aikawa for fruitful conversations on the subject presented here. His suggestions have been greatly useful for this research. I am grateful to Józef Siciak for talking with me about the Hölder continuity property of Green’s functions for domains with uniformly perfect boundary at Lublin in 1998 and for sending his paper to me later. It was quite nice for me to discuss with Stephan Gardiner about this subject when I visited Dublin in 1999. I am also thankful to Rich Stankewitz for valuable comments.

2. Local harmonic measures

Let Ω be a subdomain, or, more generally, an open subset, of the Riemann sphere. Throughout this article, to avoid the trivial case, we assume that Ω is of hyperbolic type, in other words, the boundary ∂Ω consists of at least three points unless otherwise indicated. Here and hereafter, the boundary ∂Ω and the complement Ω^{c} of Ω will be taken in the Riemann sphere ℂ̂. However, the point at infinity ∞ will play a special role in our arguments below. So, we will employ the special notation ∂_{b}Ω = ∂Ω \ {∞}.

2.1. PWB-solutions for the Dirichlet problem. First of all, we recall several notions from potential theory. The reader is referred to [17] and [11] for standard textbooks in potential theory. However, contrary to the tradition and to the above textbooks, we allow a superharmonic function to be constantly +∞ in a component of Ω. More precisely, a function s : Ω → (−∞, +∞] will be called superharmonic if s is lower semi-continuous and satisfies the condition that, for any closed Jordan domain Δ in Ω and for any harmonic function u on a neighbourhood of Δ, the condition u ≤ s on ∂Δ implies the stronger one u ≤ s on Δ. We also adopt the similar rule for subharmonic functions.

For a function φ on ∂Ω, let \overline{P}(φ) be the set of upper functions for φ in Ω, where s is said to be an upper function for φ in Ω if s is superharmonic and bounded below in Ω and satisfies

\underline{s}(ζ) := lim inf_{z→ζ in Ω} s(z) ≥ φ(ζ)

for each ζ ∈ ∂Ω. Similarly, the set \underline{P}(φ) of lower functions for φ in Ω can be defined, namely, s ∈ \underline{P}(φ) if and only if −s ∈ \overline{P}(−φ). We note that for any u ∈ \overline{P}(φ) and any v ∈ \underline{P}(φ) we have v ≤ u in Ω by the usual minimum principle for superharmonic functions. Perron’s principle says that the function

u(z) = inf_{s∈\overline{P}(φ)} s(z)

is harmonic unless u ≡ +∞ in each component of Ω. We call the function u the upper solution of (1.1) for φ in Ω and denote it by \overline{H}^{Ω}φ. Similarly, \underline{H}^{Ω}φ(z) = sup_{s∈\underline{P}(φ)} s(z) =

−\overline{H}^{Ω}(−φ)(z) is called the lower solution of (1.1). By the above notice, we see that \underline{H}^{Ω}φ ≤ \overline{H}^{Ω}φ always holds in Ω. A boundary function φ on ∂Ω is called resolutive in Ω if \overline{H}^{Ω}φ = \underline{H}^{Ω}φ, and then this common function will be denoted by H^{Ω}φ, which is called the solution of Dirichlet problem (1.1) in the sense of Perron-Wiener-Brelot, or PWB solution for φ in Ω. We make sure that the PWB solution is always a (finite-valued) harmonic function in Ω if it exists.

2.2. Regularity and barriers. A point a in ∂Ω is called a regular boundary point of Ω if H^{Ω}φ is continuously extended to φ(a) at a for each bounded resolutive function φ on ∂Ω which is continuous at a. The above regularity is known to be characterized by the existence of a barrier of Ω at a. In particular, the regularity is a local property. Here, a function b is called barrier of Ω at a if there exists an open neighbourhood V of a satisfying the following three properties:

1. b is a positive superharmonic function in Ω ∩ V,

2. lim_{z→a in Ω} b(z) = 0, and

3. inf_{z∈Ω∩V \W} b(z) > 0 for all neighbourhood W of a.

We note here Kellogg’s theorem stating that the set of irregular boundary points of a domain is polar, i.e., of capacity zero (cf. [17, Theorem 8.34]). Therefore, if a harmonic function u in Ω is the PWB solution for a bounded continuous function φ on ∂_{b}Ω, then lim_{z→ζ in Ω} u(z) = φ(ζ) for each point ζ in ∂_{b}Ω except for a polar set. In such a situation, we will conventionally say that u = φ quasi-everywhere (q.e.) in ∂_{b}Ω. Moreover, for a bounded resolutive measurable function φ, we can say that H^{Ω}φ = φ q.e. in the set of continuity of φ.

2.3. The (generalized) minimum principle. Note also the following generalization of the minimum principle for superharmonic functions (see [43, Theorem III.28] or [17, Theorem 7.10]), which will be frequently used in the sequel: Let s be a continuous superharmonic function in a domain Ω which is bounded below. If \underline{lim}_{z→ζ in Ω}s(z) ≥ 0 for each point ζ in ∂Ω except for a polar set, then s ≥ 0 in Ω. Therefore, a polar set is negligible for bounded harmonic functions. Since the point at infinity is polar in the twodimensional case, bounded solutions of the Dirichlet problem in Ω are determined by the data of the boundary function on finite boundary ∂_{b}Ω. Hence, we consider only bounded Borel (or continuous) functions on ∂_{b}Ω as boundary functions in the following.

We record here another important fact about the resolutivity of standard class of boundary functions. The following result can be seen from [11, Part 1, Chapter VIII] (the case when Ω is a bounded domain follows also from [17, Theorem 8.13]).

Theorem A. Suppose that the boundary of an open set Ω in ℂ̂ is of positive capacity. Then, all the bounded Borel functions on ∂Ω are resolutive in Ω.

2.4. Harmonic measures. For a while, we assume that a domain (or an open set) Ω has boundary of positive capacity. For a Borel subset E of ∂Ω, the harmonic measure of E relative to Ω, which will be denoted by ω(·, E, Ω), is defined as the PWB solution in Ω for the boundary value 1_{E}, where 1_{E} denotes the defining function of E. Here we note that 1_{E} is resolutive in Ω by the above theorem. As is well known, the set function E ↦ ω(z, E, Ω) for a fixed point z ∈ Ω is a Borel probability measure representing the point-evaluation of harmonic extensions at z, namely,

H^{Ω}φ(z) = ∫_{∂Ω} φ(ζ)ω(z, dζ, Ω) (2.1)

holds for every bounded Borel function φ on ∂Ω.

Let a be a finite point of the boundary of Ω. To measure the size of the boundary of Ω near the point a, the harmonic measure is convenient. A standard quantity is

ω̂_{a,r,Ω} = ω(·, ∂Ω \ B(a, r), Ω),

the harmonic measure of ∂Ω \ B(a, r) relative to Ω, where B(a, r) = {z ∈ ℂ; |z − a| ≤ r}. Intuitively, we can think that the size of ∂Ω ∩ B(a, r) is large if ω̂_{a,r,Ω}(z) tends to 0 rapidly when z approaches a. For example, we have the following estimate for a bounded continuous function φ on ∂_{b}Ω in terms of ω̂_{a,r,Ω}:

|H^{Ω}φ(z) − φ(a)| = |∫_{∂_{b}Ω}(φ(ζ) − φ(a))ω(z, dζ, Ω)|
≤2∥φ∥_{∞}ω̂_{a,r,Ω}(z) + sup_{|ζ−a|≤r} |φ(ζ) − φ(a)|(1 − ω̂_{a,r,Ω}(z)).

This simple estimate is, however, sometimes not sufficient to obtain a sharp order estimate of H_{φ}(z) − φ(a) even if we know the precise asymptotic behaviour of ω̂_{a,r,Ω}(z) as r → +0 or as z → a. Furthermore, this quantity is global, and hence, not easy to calculate directly.

2.5. Local harmonic measures. On the other hand, we will have a great advantage if we consider the local harmonic measure

ω_{a,r,Ω} := ω(·, Ω ∩ ∂B(a, r), Ω ∩ B^{∘}(a, r))

for a ∈ ∂_{b}Ω and r > 0 (usually we take r < diam ∂Ω). The subscript Ω in the above will be omitted when we do not need to express Ω explicitly. Noting that each point in ∂(Ω ∩ B^{∘}(a, r)) ∩ ∂B(a, r) is regular with respect to Ω ∩ B^{∘}(a, r) (cf. Theorem 5.2 below), we see that ω_{a,r,Ω} = 1 in Ω ∩ ∂B(a, r). On the other hand, ω_{a,r,Ω} = 0 q.e. in ∂Ω ∩ B^{∘}(a, r).

The relation between ω_{a,r,Ω} and ω̂_{a,r,Ω} is as follows:

ω̂_{a,r,Ω}(z) ≤ ω_{a,r,Ω}(z) (2.2)

for z ∈ Ω ∩ B^{∘}(a, r). Thus the local harmonic measure is stronger than the global one in this sense. To emphasize the contrast, we will call ω̂_{a,r,Ω} the global harmonic measure for a and r in this article. Inequality (2.2) is shown by applying the (generalized) minimum principle in Ω ∩ B^{∘}(a, r) (see also the proof of Lemma 5.1 below). This kind of inequality is called Carleman’s principle of domain extension (see, for instance, [15, Chap. VIII §4]). We will mention the global harmonic measures in §4.4 again.

2.6. Characterization of the boundary regularity in terms of local harmonic measures. The local harmonic measures could be good substitutions for barriers in the following sense, although ω_{a,r,Ω} does not satisfy the third condition of the barrier above.

Proposition 2.1. Let Ω be an open subset of ℂ̂ and let a ∈ ∂_{b}Ω. Then the following are equivalent.

(a) a is a regular boundary point of Ω.

(b) There exists a barrier of Ω at a.

(c) There exists a positive constant r_{0} such that ω_{a,r,Ω}(z) → 0 as z → a in Ω for any r with 0 < r < r_{0}.

Proof. The equivalence of (a) and (b) is classical. In particular, from this it follows that the regularity is a local property. Since (a)⇒(c) is clear, we give the proof only for (c) ⇒(a). The technique presented in this proof will be used in several times in the sequel. So, we explain it in detail here and we will omit the details after the second appearance.

Assume condition (c) and suppose that a bounded Borel function φ on ∂_{b}Ω is continuous at a. Without loss of generality, we may assume φ(a) = 0. Set u = H^{Ω}φ and let M = sup_{ζ∈∂_{b}Ω} |φ(ζ)|. For a given number ε > 0, we can choose a number δ with 0 < δ ≤ r_{0} such that |φ(ζ)| ≤ ε for all ζ ∈ ∂Ω with |ζ − a| ≤ δ.

Now consider the function φ_{0} = φ · 1_{B(a,δ)} and φ_{1} = φ − φ_{0} and set u_{j} = H^{Ω}φ_{j} for j = 0, 1. Since |φ_{0}| ≤ ε, we see that |u_{0}| ≤ ε in Ω. On the other hand, we have −Mω_{a,r,Ω} ≤ u_{1} ≤ Mω_{a,r,Ω} in Ω ∩ B^{∘}(a, r) for any r < δ. Indeed, u_{1} = 0 q.e. in ∂Ω ∩ B^{∘}(a, δ)(⊃ ∂Ω ∩ B(a, r)) and ω_{a,r,Ω} = 1 in Ω ∩ ∂B(a, r). Since |u_{1}| ≤ M and ω_{a,r,Ω} ≥ 0, we see lim sup_{z→ζ}(Mω_{a,r,Ω}(z) − u_{1}(z)) ≥ 0 for each ζ ∈ ∂(Ω ∩ B^{∘}(a, r)) except for a polar set. Now the (generalized) minimum principle implies Mω_{a,r,Ω} − u_{1} ≥ 0 in Ω ∩ B^{∘}(a, r). The other inequality −Mω_{a,r,Ω} ≤ u_{1} can be obtained similarly.

By condition (c), we now obtain lim_{z→a} u_{1}(z) = 0. Noting u = u_{0} + u_{1}, we therefore have

lim sup_{z→a} |u(z)| = lim sup_{z→a} |u_{0}(z)| ≤ ε.

Since ε > 0 is arbitrary, we conclude lim_{z→a} u(z) = 0 = φ(a). Now condition (a) has been shown.

2.7. LHMD property. As the reader may guess from the above proof, if ω_{a,r,Ω}(z) decay in some specific way when z tends to a the boundary behaviour of H^{Ω}φ may be controlled at a in a definite way. Indeed, we will show this in the following form.

For a point a ∈ ∂_{b}Ω and for a constant α ∈ (0, +∞), consider the condition that

(2.3) ω_{a,r,Ω}(z) ≤ C (\frac{|z − a|}{r})^{α} for all z ∈ Ω ∩ B^{∘}(a, r) and for all 0 < r < r_{0},

where C and r_{0} are positive constants. This will be called the local harmonic measure decay property or LHMD property, for short, at a with exponent α. By the above proposition, a point satisfying LHMD property is a regular boundary point. This kind of property was implicitly used for NTA(=Non-Tangentially Accessible) domains in [23, Lemma 4.1] by Jerison and Kenig, and for uniformly John domains with uniformly perfect boundary in [5, Lemma 1.1] by Balogh and Volberg.

2.8. Main result. The following is our fundamental result. In the course of preparation of this article, H. Aikawa suggested a simplification of the original proof to the author. We give here the simplified proof due to Aikawa.

Theorem 2.2. Let Ω be an open subset of the Riemann sphere with boundary of positive capacity. Suppose that a finite boundary point a of Ω satisfies LHMD property (2.3) with exponent α ∈ (0, 1] for constants C ≥ 1 and r_{0} > 0. For a constant γ ∈ (0, α), if a bounded real-valued boundary function φ satisfies that φ ≤ K on ∂_{b}Ω and that φ(ζ) ≤ L|ζ −a|^{γ} for each ζ ∈ ∂_{b}Ω ∩ B(a, r_{0}) for positive constants K and L, then the upper solution u = \overline{H}^{Ω}φ of φ satisfies the inequality

u(z) ≤ 2^{α}C (\frac{K}{r_{0}^{γ}} + \frac{L}{α − γ} ) |z − a|^{γ}, z ∈ Ω. (2.4)

Furthermore if φ = 0 on ∂Ω ∩ B(a, r_{1}) for some 0 < r_{1} ≤ r_{0}, then we have

u(z) ≤ CK (\frac{|z − a|}{r_{1}})^{α} , z ∈ Ω. (2.5)

Proof. Fix a point z_{0} in Ω and set δ_{j} = 2^{j}|z_{0} − a| for j = 0, 1, …. First we suppose that |z_{0} − a| ≤ r_{0}. Let N be the smallest integer such that δ_{N} > r_{0}. Note the inequality

2^{−N} < |z_{0} − a|/r_{0}. (2.6)

Set φ_{j} = φ · 1_{B(a,δ_{j})\B(a,δ_{j−1})} for j = 1, 2, …, N − 1, φ_{0} = φ · 1_{B(a,δ_{0})}, and φ_{N} = φ · 1_{∂Ω\B(a,δ_{N−1})}.

Since φ = φ_{0} + φ_{1} + ⋯ + φ_{N}, the subadditivity of the operator \overline{H}^{Ω} implies u ≤ u_{0} + u_{1} + ⋯ + u_{N}, where u_{j} = \overline{H}^{Ω}φ_{j}. The inequality φ_{j} ≤ Lδ_{j}^{γ} implies u_{j} ≤ Lδ_{j}^{γ} for j = 0, 1, …, N −1. In addition, we have u_{N} ≤ K. Since u_{j} = 0 q.e. in ∂Ω∩B^{∘}(a, δ_{j−1}), we have u_{j} ≤ Lδ_{j}^{γ}ω_{a,r} on Ω∩B^{∘}(a, r) for any r with |z_{0}−a| < r < δ_{j−1} and for j = 1, …, N −1 by the maximum principle. Now (2.3) yields u_{j}(z_{0}) ≤ CLδ_{j}^{γ}(δ_{0}/r)^{α}. Letting r tend to δ_{j−1}, we obtain

u_{j}(z_{0}) ≤ CLδ_{j}^{γ} (\frac{δ_{0}}{δ_{j−1}} )^{α} = 2^{α}CL2^{(γ−α)j}|z_{0} − a|^{γ}

for j = 1, …, N − 1. We note that u_{0}(z_{0}) ≤ Lδ_{0}^{γ} = L|z_{0} − a|^{γ} ≤ 2^{α}CL|z_{0} − a|^{γ} and thus the above inequality is valid also for j = 0. For j = N, by (2.6), we have the similar inequality

u_{N}(z_{0}) ≤ CK (\frac{δ_{0}}{δ_{N−1}} )^{α} = 2^{α}CK2^{−αN} < 2^{α}CK (\frac{|z_{0} − a|}{r_{0}})^{α} ≤ 2^{α}CK (\frac{|z_{0} − a|}{r_{0}})^{γ} .

By summing up these inequalities, we obtain

u(z_{0}) ≤ u_{0}(z_{0}) + ⋯ + u_{N}(z_{0}) ≤ 2^{α}C (\frac{K}{r_{0}^{γ}} + \frac{L}{1 − 2^{γ−α}} ) |z_{0} − a|^{γ}.

Secondly, if |z_{0} − a| > r_{0}, then we have u(z_{0}) ≤ K ≤ K(|z_{0} − a|/r_{0})^{γ}, in particular, the above inequality is still valid in this case. Thus we have shown the first part of the theorem.

Finally, consider the case that φ = 0 on ∂Ω ∩ B(a, r_{1}) for some 0 < r_{1} ≤ r_{0}. We may assume |z_{0}−a| < r_{1}. In the same way as above, we can see that u ≤ Kω_{a,r,Ω} in Ω∩B^{∘}(a, r) for any r with |z_{0} − a| < r < r_{1}. Using condition (2.3) and letting r → r_{1}, we obtain (2.5) for z = z_{0}.

Under the same circumstances in the above theorem, suppose that a Borel function φ on ∂_{b}Ω satisfies |φ − φ(a)| ≤ K on ∂_{b}Ω and |φ(ζ) − φ(a)| ≤ L|ζ − a|^{γ} for constants K, L and γ with 0 < γ < α. Then, by (2.4), the PWB solution u for φ in Ω satisfies the inequality

|u(z) − φ(a)| ≤ 2^{α}C (\frac{K}{r_{0}^{γ}} + \frac{L}{α − γ} ) |z − a|^{γ}, z ∈ Ω. (2.7)

In particular, we have the following.

Corollary 2.3. If a finite boundary point of a plane domain Ω satisfies the LHMD property with exponent α, then the point is γ-Hölder regular with respect to Ω for all γ with 0 < γ < α.

3. Characterizations of LHMD property

We state here characterizations of LHMD property in connection with the capacity density condition and with the Δ-regularity.

3.1. Capacity density. First, we recall the notion of (logarithmic) capacity. For a compact subset E of ℂ, the (logarithmic) capacity Cap(E) is defined to be the number e^{−c} such that

G_{D}(z, ∞) = log |z| + c + o(1)

as z → ∞, where G_{D}(·, ∞) denotes Green’s function of the (unique) unbounded connected component D of ℂ̂ \ E with pole at infinity. The constant c = − log Cap(E) is called the Robin constant of E.

For a closed set E in ℂ̂, define the (lower) capacity density at a ∈ E, denoted by CD(a, E), by

CD(a, E) = lim inf_{r→0} \frac{Cap(E ∩ B(a, r))}{r}

= lim_{r_{0}→0} CD_{r_{0}}(a, E),

where CD_{r_{0}}(a, E) = inf{Cap(E ∩ B(a, r))/r; 0 < r < r_{0}} for 0 < r_{0} ≤ +∞. Since Cap(B(a, r)) = r, we see 0 ≤ CD_{r_{0}}(a, E) ≤ CD(a, E) ≤ 1. For an open set Ω in ℂ̂, a finite boundary point a of it is said to satisfy the capacity density condition with respect to Ω if CD(a, Ω^{c}) > 0. We remark that several kinds of capacity density conditions have been considered in the literature (see, for instance, [27], [21] and [37]).

3.2. Δ-regularity. A. Ancona introduced the notion of (uniform) Δ-regularity in [3]. In this article, a finite boundary point a of an open set Ω will be called Δ-regular if there exist positive constants ε and r_{0} such that ω_{a,r,Ω} ≤ 1 − ε in Ω ∩ ∂B(a, r/2) for all 0 < r < r_{0}.

When we try to interpret the Δ-regularity at the point a as a notion of how the local harmonic measure at a decreases as we approach a, we come to see that this is nothing but the LHMD property in the following theorem. As was stated, a prototype of the next theorem was stated by A. Ancona [3], however his statement handles only the case that the dimension is greater than two.

3.3. Characterizations.

Theorem 3.1. For a fixed r_{0} with 0 < r_{0} ≤ +∞ and for a finite boundary point a of an open set Ω in ℂ̂, the following are equivalent.

(1) a satisfies the capacity density condition with respect to Ω, namely, there exists a positive constant c such that Cap(B(a, r) ∩ Ω^{c}) ≥ cr holds for any 0 < r < r_{0}.

(2) For any 0 < t < 1, there exists a constant ε ∈ (0, 1) such that ω_{a,r,Ω} ≤ 1 − ε on Ω ∩ ∂B(a, tr) for any 0 < r < r_{0}.

(3) There exist constants t, ε ∈ (0, 1) such that ω_{a,r,Ω} ≤ 1 − ε on Ω ∩ ∂B(a, tr) for any 0 < r < r_{0}.

(4) There exist positive constants C and α such that

ω_{a,r,Ω}(z) ≤ C (\frac{|z − a|}{r})^{α}

for any 0 < r < r_{0} and z ∈ B^{∘}(a, r) ∩ Ω.

Moreover, these constants depend only on each other, not on the particular point a.

Remark. In [43, p. 104], as a consequence of Wiener’s criterion, it is stated that the condition lim sup_{r→0} Cap(B(a, r) ∩ Ω^{c})/r > 0 implies the regularity of a with respect to Ω.

3.4. Some technical quantities. To make our statement brief, we define several technical quantities. Let α_{r_{0}}(a, Ω) denote the supremum of possible exponents of LHMD property at a ∈ ∂_{b}Ω with respect to Ω for 0 < r < r_{0}, in other words,

α_{r_{0}}(a, Ω) = lim_{δ→0} inf_{0<r<r_{0}} inf_{z∈Ω∩B(a,δr)} \frac{log ω_{a,r,Ω}(z)}{log(|z − a|/r)}.

By the monotonicity ω_{a,r,Ω} ≤ ω_{a,r′,Ω} in Ω∩B^{∘}(a, r′) for 0 < r′ < r (proved by the minimum principle), it is clear that this quantity does not depend on the particular choice of r_{0} > 0 if r_{0} is finite. Hence we can write simply α_{r_{0}}(a, Ω) = α(a, Ω) for 0 < r_{0} < +∞.

Furthermore, we use the more technical quantities ε_{r_{0}}(a, t, Ω) and ε(a, t, Ω) for 0 < t < 1, a ∈ ∂_{b}Ω and 0 < r_{0} ≤ +∞ which are defined by

ε_{r_{0}}(a, t, Ω) = 1 − sup_{0<r<r_{0}} sup_{z∈Ω∩B^{∘}(a,tr)} ω_{a,r,Ω}(z),

and ε(a, t, Ω) = lim_{r_{0}→0} ε_{r_{0}}(a, t, Ω). The Δ-regularity at a means that ε(a, 1/2, Ω) > 0.

By using these quantities, apart from the dependence of the constants in the statement, we can restate the above theorem as the equivalence of the following four conditions: (1’) CD_{r_{0}}(a, Ω^{c}) > 0. (2’) ε_{r_{0}}(a, t, Ω) > 0 for all 0 < t < 1. (3’) ε_{r_{0}}(a, t, Ω) > 0 for some 0 < t < 1. (4’) α_{r_{0}}(a, Ω) > 0. In the case r_{0} < +∞, as we remarked, α_{r_{0}}(a, Ω) = α(a, Ω). Therefore, these conditions

do not depend on the particular choice of r_{0} as far as r_{0} < +∞. In particular, we can eliminate r_{0} from the above four conditions.

From now on, we concentrate on the proof of the equivalence of the above four conditions. The dependence of the constants can be observed in the proof below, so we will omit the detailed discussion for it. The implication (2’)⇒(3’) is trivial.

3.5. Proof of (3’)⇔(4’). The equivalence (3’)⇔(4’) can be seen from the following lemma.

Lemma 3.2. Let 0 < r_{0} ≤ +∞. For any t ∈ (0, 1), we have the inequality

(3.1) \frac{log(1 − ε_{r_{0}}(a, t, Ω))}{log t} ≤ α_{r_{0}}(a, Ω) ≤ lim inf_{δ→0} \frac{log(1 − ε_{r_{0}}(a, δ, Ω))}{log δ}.

As an immediate consequence of the above inequality, we see that the limit exists and satisfies the identity

α_{r_{0}}(a, Ω) = lim_{t→0} \frac{log(1 − ε_{r_{0}}(a, t, Ω))}{log t}.

Proof of Lemma 3.2. Set ε = ε_{r_{0}}(a, t, Ω). Since ω_{a,r} ≤ 1 − ε on Ω ∩ ∂B(a, tr) for 0 < r < r_{0}, we can show

ω_{a,r} ≤ (1 − ε)ω_{a,tr} on Ω ∩ B^{∘}(a, tr)

by the generalized minimum principle. The repetition of this gives us the estimate

ω_{a,r} ≤ (1 − ε)^{n}ω_{a,t^{n}r} on Ω ∩ B^{∘}(a, t^{n}r)

for any 0 < r < r_{0}. Therefore, for z ∈ Ω ∩ B^{∘}(a, r) if we choose an integer n such that t^{n+1}r ≤ |z − a| < t^{n}r, then we have

ω_{a,r}(z) ≤ (1 − ε)^{n}ω_{a,t^{n}r}(z) < (1 − ε)^{n} ≤ \frac{1}{1 − ε} (\frac{|z − a|}{r})^{α} ,

where α = log(1 − ε)/ log t. It is worthwhile recording this inequality separately:

ω_{a,r,Ω}(z) ≤ \frac{1}{1 − ε_{r_{0}}(a, t, Ω)} (\frac{|z − a|}{r})^{α} , for z ∈ Ω ∩ B(a, tr), 0 < r < r_{0}, (3.2)

where

α = \frac{log(1 − ε_{r_{0}}(a, t, Ω))}{log t}. (3.3)

Hence, we obtain α_{r_{0}}(a, Ω) ≥ log(1 − ε_{r_{0}}(a, t, Ω))/ log t. Thus the left-hand side in (3.1) is proved.

In order to show the right-hand side, we take any α with 0 < α < α_{r_{0}}(a, Ω). Then, there exists a small number δ_{0} > 0 such that log ω_{a,r}(z)/ log(|z − a|/r) > α, namely, ω_{a,r}(z) < (|z − a|/r)^{α} for z ∈ Ω ∩ B(a, δr), 0 < r < r_{0} and for 0 < δ < δ_{0}. In particular, we have ε_{r_{0}}(a, δ, Ω) ≥ 1−δ^{α}, in other words, α ≤ log(1−ε_{r_{0}}(a, δ, Ω))/ log δ for 0 < δ < δ_{0}. The last inequality implies α ≤ lim inf_{δ→0} log(1 − ε_{r_{0}}(a, δ, Ω))/ log δ. Thus we get the conclusion.

3.6. Proof of (4’)⇒(1’)⇒(2’). The implications (4’)⇒(1’)⇒(2’) (and thus the completion of the proof of Theorem 3.1) follows from the arguments below. In particular, we will obtain the next result.

Proposition 3.3. Let a ∈ ∂_{b}Ω and let 0 < r_{0} ≤ +∞. Suppose that LHMD property (2.3) with exponent α > 0 holds for a constant C ≥ 1. Then, we have the inequality

− log CD_{r_{0}}(a, Ω^{c}) ≤ \frac{φ(C) + 1}{α}, (3.4)

where φ : [1, +∞) → [0, +∞) is the inverse function of x ↦ e^{x}/(1 + x). In particular, we have

α_{r_{0}}(a, Ω) ≤ \frac{φ(C) + 1}{− log CD_{r_{0}}(a, Ω^{c})}.

Conversely, for any given number m with 0 < m < 1, there exists a positive number K_{0} depending only on m such that

(3.5) \frac{m}{min\{− log CD_{r_{0}}(a, Ω^{c}), K_{0}\}} ≤ α_{r_{0}}(a, Ω).

Remark. Note that the asymptotic behaviour of φ :

φ(C) = log C + log log C + (1 + o(1))\frac{log log C}{log C} , C → ∞.

On the other hand, it is quite elementary to show the inequality cosh(3x/4) < e^{x}/(1 + x) < cosh(x) for all positive x. Therefore, we obtain the useful estimate cosh^{−1}(C) = log(C + \sqrt{C^{2} − 1}) < φ(C) < 4 cosh^{−1}(C)/3 for C > 1.

For simplicity, we assume that a = 0 and write B_{r} = B(0, r) and ω_{r} = ω_{0,r,Ω}. Denote by g_{r} Green’s function of Ω_{r} with pole at infinity, where Ω_{r} is the connected component of ℂ̂ \ (B_{r} ∩ Ω^{c}) containing ∞. Then g_{r}(z) = log |z| − log C_{r} + o(1) as z → ∞, where C_{r} = Cap(B_{r} ∩ Ω^{c}).

First we show inequality (3.4). Set ε = ε_{r_{0}}(a, t, Ω) and fix 0 < r < r_{0}. Also set M = sup_{z∈∂B_{r}∩Ω} g_{r}(z). Then, noting that g_{r} = 0 q.e. on ∂Ω∩B_{r}, we can see Mω_{r} −g_{r} ≥ 0 on Ω ∩ B_{r}^{∘} by the minimum principle. Therefore, g_{r} ≤ Mω_{r} ≤ (1 − ε)M on Ω ∩ ∂B_{tr}. In particular,

g_{r}(z) − log(|z|/tr) ≤ (1 − ε)M (3.6)

on Ω_{r} ∩ ∂B_{tr}. Since g_{r}(z) − log(|z|/tr) is harmonic near the point at infinity, we see inequality (3.6) remains valid also in Ω_{r}\B_{tr}. Using this, we have M ≤ (1−ε)M +log(1/t), which implies εM ≤ log(1/t). By (3.6) again, g_{r}(z) − log(|z|/tr) ≤ (1/ε − 1) log(1/t) for z ∈ Ω_{r} with |z| > tr. Applying this for z = ∞, we then have − log C_{r} ≤ (1/ε − 1) log(1/t) + log(1/tr) = log(1/t)/ε − log r, thus C_{r} ≥ t^{1/ε}r. The last inequality means CD_{r_{0}}(a, Ω^{c}) ≥ t^{1/ε_{r_{0}}(a,t,Ω)}, in other words,

(3.7) − log CD_{r_{0}}(a, Ω^{c}) ≤ \frac{log(1/t)}{ε_{r_{0}}(a, t, Ω)}.

Assume now condition (2.3) for α, C, r_{0}. Taking 0 < t < 1 so that Ct^{α} < 1, from inequality (3.7) we obtain

− log CD_{r_{0}}(a, Ω^{c}) ≤ \frac{log(1/t)}{1 − Ct^{α}} = \frac{x}{1 − Ce^{−αx}} =: f(x),

where we set t = e^{−x} for x > 0. Now we minimize f(x). Let x_{0} be the unique positive solution of the equation f′(x) = 0, equivalently, e^{αx} = C(1 + αx). In terms of φ defined in the statement of Proposition 3.3, we can write x_{0} = φ(C)/α. Since f(x_{0}) = x_{0} + 1/α, we get (3.4).

Secondly, we show (3.5) and implication (1’)⇒(2’). To prove these, we will use a variant of the Harnack inequality:

Lemma 3.4. For a positive constant r > 0, let u be a positive harmonic function in the domain 𝔻_{r}^{∗} := {z ∈ ℂ; |z| > r} which satisfies u(z) = log |z| + c + o(1) as z → ∞. Then u(z) > log(|z|/r) for z ∈ 𝔻_{r}^{∗}, in particular, c ≥ log 1/r. Moreover, the inequality

\frac{|z| − r}{|z| + r} ≤ \frac{u(z) − log(|z|/r)}{c + log r} ≤ \frac{|z| + r}{|z| − r}

holds for z ∈ 𝔻_{r}^{∗}.

Noting v(z) = u(z) − log(|z|/r) > 0 by the minimum principle, we apply the ordinary Harnack inequality to the positive harmonic function v(1/w) in |w| < 1/r to show the above inequality.

Now fix an arbitrary number δ with 0 < δ < t and set K = − log CD_{r_{0}}(a, Ω^{c}) ≥ 0 for some r_{0} > 0. By Lemma 3.4,

\frac{|z| − δr}{|z| + δr} ≤ \frac{g_{δr}(z) − log |z|/δr}{log δr/C_{δr}} ≤ \frac{|z| + δr}{|z| − δr},

for |z| > δr. In particular, we then have

L := min_{|z|=r} g_{δr}(z) ≥ log \frac{1}{δ} + \frac{1 − δ}{1 + δ} log \frac{δr}{C_{δr}} ,
L′ := max_{|z|=tr} g_{δr}(z) ≤ log \frac{t}{δ} + \frac{t + δ}{t − δ} log \frac{δr}{C_{δr}} .

Applying Lemma 3.4 to g_{δr} − L, we get the inequality − log C_{δr} − L + log r ≥ 0. On the other hand, noting that g_{δr} ≥ Lω_{r} on Ω ∩ B_{r}, we obtain M := sup_{z∈Ω∩∂B_{tr}} ω_{r}(z) ≤ sup_{|z|=tr} g_{δr}(z)/L = L′/L. Hence, using C_{δr} ≥ e^{−K}δr, we have

M ≤ \frac{log \frac{t}{δ} + \frac{t+δ}{t−δ} log \frac{δr}{C_{δr}}}{log \frac{1}{δ} + \frac{1−δ}{1+δ} log \frac{δr}{C_{δr}}} ≤ 1 − \frac{log \frac{1}{t} − K(\frac{t+δ}{t−δ} − \frac{1−δ}{1+δ})}{log \frac{1}{δ} + K \frac{1−δ}{1+δ}}.

Since r can be taken arbitrarily as long as 0 < r < r_{0}, we conclude

(3.8) \frac{log(1 − ε_{r_{0}}(a, t, Ω))}{log t} ≥ \frac{ε_{r_{0}}(a, t, Ω)}{log \frac{1}{t}} ≥ \frac{1 − K(\frac{t+δ}{t−δ} − \frac{1−δ}{1+δ})/ log \frac{1}{t}}{log \frac{1}{δ} + K \frac{1−δ}{1+δ}}

for any pair (t, δ) with 0 < δ < t < 1. We can easily see that, for a fixed t, the right-most term in (3.8) is positive for sufficiently small δ, and thus, we have proved the implication (1’)⇒(2’).

In order to get inequality (3.5) we choose δ and t so that t = min{(CD_{r_{0}}(a, Ω^{c}))^{β}, x} and δ = t^{2}, where the positive numbers x and β will be specified later. Set K′ = log(1/t)/β = max{K, log(1/x)/β}. By (3.8), we now have

\frac{log(1 − ε_{r_{0}}(a, t, Ω))}{log \frac{1}{t}} ≥ \frac{1 − K(\frac{1+t}{1−t} − \frac{1−t^{2}}{1+t^{2}} )/βK′}{2βK′ + K \frac{1−t^{2}}{1+t^{2}}} ≥ \frac{1 − (\frac{1+t}{1−t} − \frac{1−t^{2}}{1+t^{2}} )/β}{(2β + \frac{1−t^{2}}{1+t^{2}} )K′}

= \frac{h_{β}(t)}{K′} ≥ \frac{h_{β}(x)}{K′},

where

h_{β}(x) = \frac{1 + x^{2} − 2x(1 + x)/β(1 − x)}{2β(1 + x^{2}) + 1 − x^{2}}.

The last inequality follows from the fact that the function h_{β}(x) is monotonically decreasing in 0 < x < 1. Actually, a direct calculation yields h′_{β}(x) = −2(1 + x)^{2}(2β(1 − x + x^{2}) + 1 − x^{2})/(1 − x)^{2}(2β(1 + x^{2}) + 1 − x^{2})^{2} < 0. Since lim_{β→0} lim_{x→0} h_{β}(x) = 1, we can choose sufficiently small x > 0 and β > 0 so that h_{β}(x) > m. By Lemma 3.2, if we set K_{0} = log(1/x)/β, we have (3.5).

Remark. It is not difficult to obtain practically an explicit value K_{0} for a given m between 0 and 1. For example, suppose we are given m = 1/4. Then, if we choose β = 1 and x = 0.1, we have h_{1}(0.1) = 0.254 ⋯ > 1/4. Therefore, we can take K_{0} = − log 0.1 ≈ 2.3.

4. LHMD property and uniform perfectness

4.1. Global and uniform capacity density conditions. Ch. Pommerenke [32] proved that a compact set E in the Riemann sphere ℂ̂ is uniformly perfect if and only if E satisfies the global capacity density condition: CD_{d}(E) > 0, where d = diam E and we write CD_{r_{0}}(E) = inf_{a∈E\{∞}} CD_{r_{0}}(a, E) for 0 < r_{0} ≤ +∞. Moreover, he gave the explicit estimate

k(E)^{2}/32 ≤ CD_{d}(E) ≤ k(E), (4.1)

where k(E) is the supremum of numbers c ≥ 0 so that E ∩ {z; cr ≤ |z − a| ≤ r} ≠ ∅ holds for any a ∈ E \ {∞} and 0 < r < diam E.

We will say that E satisfies the uniform capacity density condition if CD(E) > 0, where CD(E) = lim_{r_{0}→0} CD_{r_{0}}(E). When diam E < +∞, the global capacity density condition is same as uniform capacity density condition. Note that this is not true in general if diam E = +∞.

4.2. Global and uniform LHMD properties. In an analogous way, we introduce two kinds of LHMD properties for open sets. An open set Ω will be said to satisfy the uniform LHMD property with exponent α > 0 if there exist constants 1 ≤ C < +∞ and 0 < r_{0} ≤ +∞ such that ω_{a,r,Ω}(z) ≤ C(|z − a|/r)^{α} for all r with 0 < r < r_{0}, a ∈ ∂_{b}Ω and z ∈ Ω ∩ B(a, r). Furthermore if we can take r_{0} = d = diam ∂Ω, we will say that Ω satisfies the global LHMD property. Note that the global LHMD property coincides with the uniform LHMD property if d < +∞ (cf. §4.3).

By this terminology, we can say that the complement of an open subset Ω of ℂ̂ with #∂Ω ≥ 2 is uniformly perfect if and only if Ω satisfies the global LHMD property. In particular, the global LHMD property is preserved by quasiconformal mappings on Ω although the exponent may change, because the uniform perfectness is invariant under quasiconformal mappings (see, for instance, [41]).

4.3. Example. In order to illustrate the need to distinguish the term “global LHMD” from “uniform LHMD”, we give a simple example. Let {a_{n}} be a sequence of real numbers satisfying the condition a_{n+1} − a_{n} → +∞ as n → ∞. Then, the domain Ω = ℂ \ ⋃_{n=1}^{∞} B(a_{n}, 1) satisfies the uniform LHMD property with exponent 1 for any positive finite number r_{0} > 0 (see Theorem 5.2 below), but not for r_{0} = +∞ because Ω^{c} is not uniformly perfect.

4.4. Relation between local and global harmonic measures. We mention here an important relation between local and global harmonic measures. In view of (2.2), if a boundary point a satisfies LHMD property (2.3) with exponent α, then the global harmonic measure ω̂_{a,r,Ω} satisfies the same inequality. It is a remarkable fact that the converse is also true in general. For a bounded domain in ℝ^{n} (n ≥ 2), H. Aikawa [2] proved the following: Suppose that the global harmonic measures satisfy ω̂_{a,r,Ω}(z) ≤ C(|z−a|/r)^{α} for all a ∈ ∂_{b}Ω, r with 0 < r < r_{0} and z ∈ Ω ∩ B^{∘}(a, r), where α, C and r_{0} are positive constants. Then ω_{a,r,Ω}(z) ≤ C′(|z − a|)^{α} holds for all a ∈ ∂_{b}Ω, r with 0 < r < r′_{0} and z ∈ Ω ∩ B^{∘}(a, r), where C′ and r′_{0} are constants.

4.5. Characterization of uniform perfectness in terms of hyperbolic metric. The uniform perfectness can be characterized also in terms of the hyperbolic metric. Let Ω be an open set in ℂ̂ of hyperbolic type, namely, #∂Ω ≥ 3. Then the hyperbolic metric λ_{Ω}(z)|dz| of curvature −4 can be defined on each connected component of Ω. Let δ_{Ω} be the distance function of Ω, namely δ_{Ω}(z) = dist(z, ∂_{b}Ω) = min{|z − a|; a ∈ ∂_{b}Ω}. For 0 < r ≤ +∞, we consider the open subset Ω(r) = {z ∈ Ω; δ_{Ω}(z) < r}. The following result is essentially due to Beardon and Pommerenke [6]. (In the case when ∞ ∈ Ω, we need something more but we omit it.)

Theorem B. Let Ω be an open subset of ℂ̂ of hyperbolic type. Then the complement Ω^{c} is uniformly perfect if and only if

inf_{z∈Ω(d)} δ_{Ω}(z)λ_{Ω}(z) > 0,

where d = diam ∂Ω.

4.6. Localized quantities measuring boundary. In view of the relation

inf_{z∈Ω(d)} δ_{Ω}(z)λ_{Ω}(z) = inf_{z∈Ω(d)} inf_{a∈∂_{b}Ω} |z − a|λ_{Ω}(z) = inf_{a∈∂_{b}Ω} inf_{z∈Ω(d)} |z − a|λ_{Ω}(z),

it may be natural to introduce the quantities

μ_{r}(a, Ω) = inf_{z∈Ω∩B(a,r)} |z − a|λ_{Ω}(z) and ν_{r}(a, Ω) = inf_{z∈Ω∩B(a,r)} δ_{Ω}(z)λ_{Ω}(z)

for 0 < r ≤ +∞ and a ∈ ℂ \ Ω. We conventionally define these values to be +∞ if Ω ∩ B(a, r) is empty. Note that Zheng [44] introduced a similar quantity (which is, in fact, same as μ_{∞}(a, Ω)) and proved several related results. Since δ_{Ω}(z) ≤ |z − a|, we always have ν_{r}(a, Ω) ≤ μ_{r}(a, Ω). Then, actually, one can easily check

inf_{z∈Ω(r)} δ_{Ω}(z)λ_{Ω}(z) = inf_{a∈∂_{b}Ω} μ_{r}(a, Ω) = inf_{a∈∂_{b}Ω} ν_{r}(a, Ω)

for any 0 < r ≤ +∞. Letting r = diam ∂Ω, we can summarize these observations as follows.

Theorem 4.1. Let Ω be an open set in ℂ̂ with #∂Ω ≥ 3 and set d = diam ∂Ω. Then, the following conditions are equivalent.

(a) ∂Ω is uniformly perfect.

(b) Ω satisfies global LHMD property. In other words, there exist constants C ≥ 1 and α > 0 such that ω_{a,r,Ω}(z) ≤ C(|z − a|/r)^{α} for all a ∈ ∂_{b}Ω, 0 < r < d and z ∈ Ω ∩ B^{∘}(a, r).

(c) inf{CD_{d}(a, Ω^{c}); a ∈ ∂_{b}Ω} > 0.

(d) inf{μ_{d}(a, Ω); a ∈ ∂_{b}Ω} > 0.

(e) inf{ν_{d}(a, Ω); a ∈ ∂_{b}Ω} > 0. Moreover, if d < +∞, we can replace d by any positive number r_{0} in the above conditions.

4.7. A geometric localized quantity. Now we further introduce a geometric quantity. We say that A is a round annulus centered at a if A has the form {z ∈ ℂ; r_{1} < |z−a| < r_{2}} for constants 0 ≤ r_{1} < r_{2} ≤ +∞ and set M(A) = \sqrt{r_{1}r_{2}} and mod(A) = log(r_{2}/r_{1}). Let a ∈ ℂ \ Ω. For 0 < r ≤ +∞, we denote by A_{r}(a, Ω) the set of round annuli A in Ω centered at a with M(A) ≤ r. Note that, if 2r ≤ d = diam ∂Ω, each annulus in A_{r}(a, Ω) separates ∂Ω. Finally, set

m_{r}(a, Ω) = sup_{A∈A_{r}(a,Ω)} mod(A).

Here, we define m_{r}(a, Ω) = 0 if A_{r}(a, Ω) is empty. We note the obvious relation

(4.2) − log k(Ω^{c}) = sup_{a∈ℂ\Ω}m_{d}(a, Ω),

where d = diam ∂Ω.

Using the inequality due to J. A. Hempel [18] and J. A. Jenkins [22]:

λ_{D_{a,b}}(z) ≥ \frac{1}{|z − a|(2| log |\frac{z−a}{b−a}|| + K)}, (4.3)

where D_{a,b} = ℂ \ {a, b} and K = Γ(1/4)^{4}/2π^{2} ≈ 8.7537, we can show the following result.

Proposition 4.2. Let a ∈ ℂ \ Ω and 0 < r ≤ +∞. Then

\frac{1}{m_{r}(a, Ω) + K} ≤ μ_{r}(a, Ω) ≤ \frac{π}{2m_{r}(a, Ω)}.

Proof. The proof can proceed as in [6]. Let A ∈ A_{r}(a, Ω) and take a point z_{0} ∈ A so that |z_{0} − a| = M(A)(≤ r). Since λ_{Ω}(z_{0}) ≤ λ_{A}(z_{0}) = π/2mod(A)M(A) = π/2|z_{0} − a|mod(A) (see [6]), we see μ_{r}(a, Ω) ≤ |z_{0} − a|λ_{Ω}(z_{0}) ≤ π/2mod(A). Thus, the right-hand side has been obtained.

Next, we prove the left-hand side. Take any number m with m > m_{r}(a, Ω). Fix an arbitrary point z_{0} in Ω∩B(a, r) and set r_{0} = |z_{0}−a|. Then, the annulus A = {z; e^{−m/2}r_{0} < |z − a| < e^{m/2}r_{0}} must not belong to A_{r}(a, Ω), so we can find a point b ∈ ∂_{b}Ω ∩ A. Since e^{−m/2} < |(z_{0}−a)/(b−a)| < e^{m/2}, by (4.3), we have λ_{Ω}(z_{0}) ≥ λ_{D_{a,b}}(z_{0}) ≥ 1/|z_{0}−a|(m+K). Now the conclusion easily follows.

Corollary 4.3. Let a ∈ ∂_{b}Ω. Then

lim inf_{z→a in Ω} |z − a|λ_{Ω}(z) > 0 ⇔ lim sup_{r→0} m_{r}(a, Ω) < +∞.

4.8. Comparisons between localized quantities. As was seen before, the LHMD property of Ω at a can be characterized by the condition CD(a, Ω) > 0. Occasionally, a geometric characterization of it would be more preferable for us. In view of Theorem 4.1, one may expect that CD_{r}(a, Ω^{c}) > 0 is equivalent to μ_{r}(a, Ω) > 0 or to ν_{r}(a, Ω) > 0. Unfortunately, neither is true in general as we shall see later. However, these conditions are good geometric criteria to test the validity of the inequality CD_{r}(a, Ω^{c}) > 0.

Theorem 4.4. Let a ∈ ∂_{b}Ω and 0 < r ≤ diam ∂Ω. Then we have

\frac{1}{μ_{r}(a, Ω)} − \frac{Γ(1/4)^{4}}{2π^{2}} ≤ − log CD_{r}(a, Ω^{c}) ≤ \frac{π}{ν_{r}(a, Ω)} + 5 log 2

Proof. Obviously, − log CD_{r}(a, Ω^{c}) ≤ m_{r}(a, Ω)/2, so we get the left-hand side inequality from Proposition 4.2. Next, we prove the right-hand side. Set ν = ν_{r}(a, Ω). First, note the inequality ν ≤ 1/2. Actually, if we choose z_{0} in Ω∩B(a, r/2), then there exists a point b ∈ ∂Ω∩B(a, r) such that r_{0} := δ_{Ω}(z_{0}) = |z_{0}−b|. Note that the disk Δ = {z ∈ ℂ; |z−z_{0}| < r_{0}} is contained in Ω. Set z_{t} = z_{0}+t(b−z_{0}) for each t ∈ [0, 1). Since, δ_{Ω}(z_{t}) = |z_{t}−b| = (1−t)r_{0}, we obtain δ_{Ω}(z_{t})λ_{Ω}(z_{t}) ≤ δ_{Ω}(z_{t})λ_{Δ}(z_{t}) = (1 − t)r_{0} · 1/r_{0}(1 − t^{2}) = 1/(1 + t). Letting t → 1, we have ν ≤ 1/2.

Consider the open subset Ω_{0} = Ω ∩ B^{∘}(a, r) of Ω. We show the inequality

inf_{z∈Ω_{0}} δ_{Ω_{0}}(z)λ_{Ω_{0}}(z) ≥ ν.

Clearly, δ_{Ω_{0}}(z) = min{δ_{Ω}(z), r−|z−a|} holds for z ∈ Ω_{0}. Since δ_{Ω}(z)λ_{Ω_{0}}(z) ≥ δ_{Ω}(z)λ_{Ω}(z) ≥ ν, and since (r − |z − a|)λ_{Ω_{0}}(z) ≥ (r − |z − a|)λ_{B^{∘}(a,r)}(z) = r/(r + |z − a|) > 1/2 ≥ ν, we obtain the desired inequality.

Set d_{0} = diam Ω_{0} ≤ 2r. From Proposition 4.2, we deduce ν ≤ ν_{d_{0}}(b, Ω_{0}) ≤ μ_{d_{0}}(b, Ω_{0}) ≤ π/2m_{d_{0}}(b, Ω_{0}) for each b ∈ ℂ \ Ω_{0}. Noting the relation (4.2), we get − log k(Ω_{0}^{c}) ≤ π/2ν. Now we can use (4.1) to prove − log CD_{∞}(Ω_{0}^{c}) ≤ log(32/k(Ω_{0}^{c})^{2}) ≤ π/ν + 5 log 2. Since CD_{r}(a, Ω^{c}) ≥ CD_{∞}(a, Ω_{0}^{c}), we conclude the final inequality.

As a direct consequence of the above theorem, we have the following implications:

lim_{r→0} ν_{r}(a, Ω) = lim inf_{z→a in Ω} δ_{Ω}(z)λ_{Ω}(z) > 0
⇒ CD(a, Ω^{c}) = lim inf_{r→0} \frac{Cap(Ω^{c} ∩ B(a, r))}{r} > 0
⇒ lim_{r→0} μ_{r}(a, Ω) = lim inf_{z→a in Ω} |z − a|λ_{Ω}(z) > 0.

However, as we announced, these implications are both strict. We end this section with showing it by simple examples.

Example 4.1. Let ℍ denote the right half plane {z ∈ ℂ; Re z > 0} and set Ω = ℍ \ {2^{n}; n ∈ ℤ}, where ℤ stands for the set of integers. Since each 2^{n} is an isolated boundary point of Ω, we can see that δ_{Ω}(z)λ_{Ω}(z) → 0 as z → 2^{n}. Therefore, lim inf_{z→0} δ_{Ω}(z)λ_{Ω}(z) = 0.

On the other hand, since a countable set is polar, we obtain Cap(Ω^{c} ∩ B(0, r)) = Cap(ℍ^{c} ∩ B(0, r)) = C_{0}r for r > 0, where C_{0} = Cap(ℍ^{c} ∩ B(0, 1)) > 0. Hence, we have CD(0, Ω^{c}) = C_{0} > 0. This example shows that the first implication is strict.

Example 4.2. Next consider the domain Ω = ℂ^{∗} \ {2^{n}; n ∈ ℤ}, where ℂ^{∗} = ℂ \ {0}. Then, m_{r}(0, Ω) = log 2 for all 0 < r < +∞. Therefore, by Corollary 4.3, we see lim inf_{z→0} |z|λ_{Ω}(z) > 0.

On the other hand, clearly, we have CD(0, Ω^{c}) = 0. Hence, this example shows that the second implication is strict.

We remark that, in the above examples, we can replace each singleton {2^{n}} by a sufficiently small closed disk B(2^{n}, r_{n}) to obtain a domain without isolated boundary point.

5. Estimates of local harmonic measures

In the last section, we investigated geometric and analytic characterizations of the LHMD property. In particular, we saw that a domain has the global LHMD property if and only if the boundary of the domain is uniformly perfect. The exponent of LHMD is, however, not very explicit with respect to the uniform perfectness constant, or at least, it is not easy to calculate it accurately.

So, in this section, for a given exponent α > 0, we present several geometric conditions which guarantee the LHMD property with the exponent α at a boundary point.

In our argument, it is crucial to use some monotonicity principles for harmonic measures, one of which is the following.

Lemma 5.1. Suppose that domains Ω and Ω̃ satisfy Ω ⊂ Ω̃ and a ∈ ℂ \ Ω̃. Then, we have ω_{a,r,Ω} ≤ ω_{a,r,Ω̃} on Ω ∩ B^{∘}(a, r).

This can be easily verified by the generalized minimum principle.

5.1. Exterior circle condition. We start with a sufficient condition for local harmonic measures to have linear growth.

In this article, for a number ρ > 0, a domain Ω will be said to satisfy the exterior circle condition with radius ρ at a finite boundary point a of Ω, if there exists a closed disk Δ with radius ρ satisfying a ∈ ∂Δ and Δ∩Ω = ∅. This kind of condition was first considered by H. Poincaré.

If Ω satisfies the exterior circle condition with fixed radius ρ at every finite boundary point, then Ω is said to satisfy the uniform exterior circle condition with radius ρ. A domain Ω is convex if and only if Ω satisfies the exterior circle condition with arbitrarily large radius ρ > 0.

Theorem 5.2. Let Ω be a domain in the Riemann sphere satisfying the exterior circle condition with radius ρ at a ∈ ∂_{b}Ω. Then the inequality

(5.1) ω_{a,r,Ω}(z) ≤ arcsin (\sqrt{\frac{1−(r/2ρ)^{2}}{1+(|z−a|/r)^{2}+|z−a|/ρ}} · \frac{|z−a|}{r}) ; arcsin \sqrt{1 − (r/2ρ)^{2}} ≤ 2\frac{|z − a|}{r}

holds for any 0 < r < 2ρ and any point z ∈ Ω with |z − a| < r.

Proof. Without loss of generality, we may assume that a = 0, r = 1 and Δ = {|z−ρ| ≤ ρ}, and hence 1 < 2ρ. Let ζ_{0} = e^{iφ} be the intersection point of the unit circle and ∂Δ with positive imaginary part (so 0 < φ < π/2). The angle φ is determined by the relation cos φ = 1/2ρ. We set Ω̃ = ℂ̂ \ Δ and write ω = ω_{0,1,Ω} and ω̃ = ω_{0,1,Ω̃}. Since the function [(1 − ζ_{0}z)/(1 − ζ̄_{0}z)]^{π/φ} maps Ω̃ ∩ B^{∘}(0, 1) conformally onto the upper half plane, we can compute ω̃ directly as

ω̃(z) = \frac{1}{φ} arg (\frac{1 − ζ_{0}z}{1 − ζ̄_{0}z} ) = \frac{1}{φ} \{2φ − arg (\frac{ζ_{0} − z}{ζ̄_{0} − z} )\} .

If |z| = t < 1, then we have ω̃(z) ≤ ω̃(−t) = 2(φ − arg(ζ_{0} + t))/φ, because the equipotential curves of ω̃ consists of those open circular arcs in Ω̃ ∩ B^{∘}(0, 1) whose endpoints are ζ_{0} and ζ̄_{0}. Setting θ = φ − arg(ζ_{0} + t), by the sine formula, we obtain

\frac{t}{sin θ} = \frac{|ζ_{0} + t|}{sin φ} = \frac{\sqrt{1 + t^{2} + t/ρ}}{sin φ} ≥ \frac{1}{sin φ}.

Hence,

ω̃(z) = \frac{2θ}{φ} = \frac{2}{φ} arcsin (\sqrt{\frac{1 − 1/4ρ^{2}}{1 + |z|^{2} + |z|/ρ}}|z|) ≤ \frac{2}{φ} arcsin(|z| sin φ) ≤ 2|z|.

Finally, applying Lemma 5.1, we get the conclusion.

As we see from the proof, the first inequality in (5.1) is sharp.

5.2. Domains with smooth boundary of curvature bounded below. A bounded domain whose boundary is of class C^{2}, of course, satisfies the uniform exterior circle condition with sufficiently small constant ρ. However, in order to get an explicit estimate, we should impose on the curvature bound for the boundary curves. Now we formulate one of the results in this direction.

For an oriented regular C^{2} simple closed plane curve ξ : I → ℂ parametrized by the arclength, we can define the curvature κ : I → ℝ by

κ(t) = Im \frac{ξ′′(t)}{ξ′(t)} = \frac{d}{dt} arg ξ′(t).

As elementary differential geometry tells us, the curvature κ(t) represents the (signed) reciprocal of curvature radius of the curve ξ at the point ξ(t). If the curvature is nonnegative, then the domain bounded by the curve is convex. In particular, if the curvature is bounded by a positive constant from below, the domain is strictly convex.

Let Ω ⊂ ℂ̂ be a domain whose boundary in ℂ consists of finitely many disjoint smooth regular simple curves of class C^{2}. We will always assume that the inclusion of each curve into ℂ is a proper map. Therefore, if a boundary curve is not closed, it must terminate at the point at infinity in both directions. Note that we do not assume any differentiability of boundary curves at the point at infinity and admit the situation that several boundary curves meet at the point at infinity.

Here and hereafter, the boundary curves will be positively oriented with respect to the domain Ω, in other words, if we proceed along the boundary curve according to the orientation, we can see the domain Ω in the left. The curvature will be defined in accordance with this orientation.

We denote by p_{Ω} the inner distance on Ω, precisely, p_{Ω}(z, w) = inf_{ξ} diam ξ, where the infimum is taken over all paths ξ connecting z and w in Ω and diam ξ denotes the Euclidean diameter of the curve ξ. Because Ω has smooth boundary, p_{Ω} can be extended continuously on \overline{Ω} \ \{∞\}. Note that |z − w| ≤ p_{Ω}(z, w) by definition.

Now we are ready to state our theorem.

Theorem 5.3. Let Ω ⊂ ℂ̂ be a domain whose boundary in ℂ consists of smooth curves of class C^{2} with curvature not less than −1/ρ, where ρ is a positive constant or +∞. Then, for each finite boundary point a of Ω, estimate (5.1) holds for any 0 < r < 2ρ and any point z ∈ Ω with p_{Ω}(a, z) ≤ r.

The assertion above can be shown by the same method as in the proof of Theorem 5.2 if we know the following global property of plane curves.

Proposition 5.4. Let ξ be an oriented regular simple closed plane curve of class C^{2} whose curvature κ satisfies κ ≥ −1/ρ at every point on ξ, where ρ is a positive constant. For each point a on ξ, let Δ denote the open disk of radius ρ which is tangent to and sitting on the right of the curve ξ at the point a. Suppose that a subarc ξ_{1} of ξ starting from the point a intersects Δ. Then the Euclidean diameter of ξ_{1} is at least 2ρ.

This proposition is intuitively trivial, however it seems not to be easy to give a rigorous proof of it. We could not find a reference for this result, so we will include a proof of this proposition in Appendix for the convenience of the reader.

Remark. As is seen from the proof, the assumption on ξ can be weakened in the above proposition as follows: ξ is an oriented regular simple closed plane curve of class C^{1} for which ξ′ is absolutely continuous and κ = Im (ξ′′/ξ′) ≥ −1/ρ a.e., where ξ is parametrized by arclength. In particular, if ξ is of class C^{1,1}, i.e., ξ is of class C^{1} and ξ′ is Lipschitz continuous, then the above proposition is applicable.

5.3. Generalized exterior wedge condition. Our next theorem treats domains with a generalized exterior wedge condition. For the precise statement, we prepare some notation. Let Ω be an open set in the Riemann sphere. For a ∈ ℂ and 0 < r < ∞, we set

L_{r}(a, Ω) = \begin{cases}−∞ & if ∂B(a, r) ⊂ Ω, \\ |∂B(a, r) ∩ Ω^{c}|/r & otherwise, \end{cases}

here | · | denotes the one-dimensional Lebesgue measure. In particular, 0 ≤ L_{r}(a, Ω) ≤ 2π or L_{r}(a, Ω) = −∞.

For β ∈ [0, 2π), an open set Ω such that L_{r}(a, Ω) ≥ β for a ∈ ∂_{b}Ω and 0 < r < ρ, where ρ is a positive constant, will be said to satisfy the generalized exterior wedge condition at a with opening β (and with height ρ). This notion is, of course, a generalization of the usual exterior wedge condition, which means that there is a (closed) wedge of opening β and height ρ with vertex at a which lies in the complement of the domain.

For β ∈ [0, 2π), if there exists a positive constant ρ such that L_{r}(a, Ω) ≥ β for any a ∈ ∂_{b}Ω and for any 0 < r < ρ, the domain Ω is said to satisfy the uniform generalized exterior wedge condition of opening β (with height ρ). Actually, it is clear that there does not exist a domain satisfying the uniform generalized exterior wedge condition of opening β with β > π.

Theorem 5.5. Let Ω be an open subset in ℂ̂ and let β ∈ [0, 2π) and ρ > 0 be constants. Suppose that L_{r}(a, Ω) ≥ β for a finite boundary point a of Ω and for all r with 0 < r < ρ. Then the estimate

ω_{a,r,Ω}(z) ≤ \frac{4}{π} arctan ((\frac{|z − a|}{r})^{\frac{π}{2π−β}} ) < \frac{4}{π} (\frac{|z − a|}{r})^{\frac{π}{2π−β}} (5.2)

is valid for any 0 < r < ρ and z ∈ Ω ∩ B^{∘}(a, r). In particular, Ω satisfies the LHMD property with exponent π/(2π − β) at the point a.

When the domain satisfies the (usual) exterior wedge condition, the above theorem agrees with a special case of the result of K. Miller [31] concerning the boundary Hölder regularity.

When the boundary component of Ω containing a is non-degenerate, namely, it is a continuum, then Ω satisfies the generalized exterior wedge condition at a with opening 0. Hence, by the above theorem, Ω enjoys the LHMD property with exponent 1/2 at a. Combining with Theorem 2.2, we can see that a is an α-Hölder regular boundary point for any α with 0 < α < 1/2. This agrees with a result of A. Hinkkanen [19]. More specifically, if β = 0, a = 0 and r = 1, the first inequality in (5.2) takes the form

ω_{0,1,Ω}(z) ≤ \frac{4}{π} arctan (\sqrt{|z|}) = \frac{2}{π} arccos (\frac{1 − |z|}{1 + |z|}) .

This inequality is known in this case, namely, when ∂Ω contains a continuum connecting 0 with the unit circle ([36], see also [15, Chap. VIII §4]).

Proof. We may assume that a = 0 and r = 1. Let Ω^{∗} denote the circular symmetrization of Ω, precisely, Ω^{∗} = {te^{iθ}; 0 < t < ∞, 2|θ| < 2π −L_{t}(a, Ω)}. The following is a deep result due to A. Baernstein II.

Lemma 5.6 (Baernstein II [4, Theorem 7]). ω_{0,1,Ω}(z) ≤ ω_{0,1,Ω^{∗}}(|z|) for z ∈ Ω ∩ B^{∘}(0, 1).

By assumption, we know Ω^{∗}∩B^{∘}(0, 1) ⊂ Ω̃∩B^{∘}(0, 1), where Ω̃ = {te^{iθ}; 0 < t < 1, 2|θ| < 2π − β}. The function w = z^{π/(2π−β)} maps Ω̃ ∩ B^{∘}(0, 1) conformally onto the right half unit disk. Therefore, we can see

ω_{0,1,Ω̃}(z) = \frac{2}{π} \{π − arg (\frac{−i − w}{i − w})\} .

In particular, ω_{0,1,Ω̃}(|z|) = \frac{4}{π}(\frac{π}{2} − arctan \frac{1}{|w|}) = \frac{4}{π} arctan |w| = \frac{4}{π} arctan(|z|^{π/(2π−β)}). Now, combining Lemma 5.6 with Lemma 5.1, we obtain ω_{0,1,Ω}(z) ≤ ω_{0,1,Ω^{∗}}(|z|) ≤ ω_{0,1,Ω̃}(|z|), and thus we have shown the desired estimate.

5.4. Hölder regularity at outward pointing cusps. From the above result, we know the existence of domains which admit α-Hölder regular boundary points for α ≥ 1. For example, consider the sector Ω = {z ∈ ℂ^{∗}; 0 < arg z < θ} for θ ∈ (0, π). Then the origin is α-Hölder regular with respect to Ω for any α < π/θ by Theorem 2.2. Of course, such points are not generic in the boundary.

Moreover, we can observe that an outward pointing cusp is α-Hölder regular for any positive α.

Theorem 5.7. Let a be a finite boundary point of an open set Ω in ℂ̂. Suppose that L_{r}(a, Ω) ≥ 2π −r/ρ for all 0 < r < r_{0}, where ρ and r_{0} are positive constants with r_{0} ≤ 2ρ. Then, we have

ω_{a,r,Ω}(z) < \frac{4}{π} exp (\frac{πρ(|z − a| − r)}{r|z − a|}) < \frac{4}{π} exp (\frac{πρ}{r_{0}} (1 − \frac{r}{|z − a|})) (5.3)

for 0 < r < r_{0} and for z ∈ Ω ∩ B^{∘}(a, r). In particular, the point a is α-Hölder regular for any positive number α.

Proof. The proof is similar to that of Theorem 5.5. We may assume that a = 0. Let Ω^{∗} be the circular symmetrization of Ω described in the proof of Theorem 5.5. Since L_{r}(a, Ω) ≥ 2π − 2 arcsin(r/2ρ) for 0 < r < r_{0}, we obtain Ω^{∗} ⊂ Ω̃ = {re^{iθ}; 0 < r < r_{0}, |θ| < arcsin(r/2ρ)}. Geometrically, Ω̃ is the right component of the open set B^{∘}(0, r_{0})\ (B(iρ, ρ) ∪ B(−iρ, ρ)).

Now fix r with 0 < r < r_{0}. Consider the domain Ω̃_{r} = Ω̃ ∩ B^{∘}(r/2, r/2) and set u = ω(·, ∂Ω̃_{r}∩∂B(r/2, r/2), Ω̃_{r}). Then, by the minimum principle, we can see that ω_{0,r,Ω̃} < u in Ω̃_{r}. Therefore, using Lemmas 5.1 and 5.6, we obtain ω_{0,r,Ω}(z) ≤ ω_{0,r,Ω^{∗}}(|z|) ≤ ω_{0,r,Ω̃}(|z|) < u(|z|) for any z ∈ Ω ∩ B^{∘}(0, r).

We next compute the value u(x) for positive x. Note that the Möbius transformation z ↦ πρ(r − z)/rz maps Ω̃_{r} onto the domain {w ∈ ℂ; Re w > 0, |Im w| < π/2}. Hence, the function f(z) = exp(πρ(r − z)/rz) maps Ω̃_{r} conformally onto the domain W = {w ∈ ℂ; Re w > 0, |w| > 1} in such a way that f(∂Ω̃_{r} ∩ ∂B(r/2, r/2)) = {ζ; |ζ| = 1, Re ζ > 0} =: I. Since ω(y, I, W) = 2(arg(y + i) − arg(y − i))/π = 4 arctan(1/y)/π < 4/πy for y > 1, we obtain u(x) < 4/πf(x) = 4 exp(−πρ(r − x)/rx)/π. Taking x = |z|, we get the desired inequality.

As we can read from the above proof, the following geometric form also holds. Suppose that, for a ∈ ∂_{b}Ω, there exist two closed disks B_{1}, B_{2} of the same radius ρ in the outside of Ω satisfying B_{1} ∩ B_{2} = {a}. Then, the same inequality as (5.3) holds for r_{0} = 2ρ.

6. Globalization

In the previous sections, we mainly investigated the local behaviour of bounded harmonic functions at a boundary point. In this section, we use results proved there to obtain global Hölder continuity of Green’s function and of PWB solutions of Dirichlet problems under the assumption of a uniform property of the boundary.

6.1. Reduction of global Hölder continuity of Green’s function. Let Ω be a domain with boundary of positive capacity such that ∞ ∈ Ω ⊂ ℂ̂. Then, Ω admits Green’s function with pole at infinity, which will be denoted by G = G_{Ω}. Since G(z) − log |z| is harmonic and log |z| is Lipschitz continuous near the point at infinity, we only have to consider the Hölder continuity of G near the boundary of Ω. So, we may restrict our attention to the subset Ω(r_{0}) = {z ∈ Ω; δ_{Ω}(z) < r_{0}} of Ω, where r_{0} is a fixed positive number. Of course, the following results are still valid for Green’s function with pole at an arbitrary point under a suitable modification of the statement near the pole.

The first assertion enables us to reduce the problem involving two variables to that of one variable. A similar, more general result can be found, for example, in [19, Lemma A].

Lemma 6.1. Let Ω be a domain containing ∞ in ℂ̂ with regular boundary in the sense of Dirichlet. For constants α ∈ (0, 1] and r_{0} > 0, the following conditions are equivalent:

(i) G_{Ω} is Hölder continuous with exponent α in Ω(r_{0}), namely, there exists a constant

C_{1} such that

|G_{Ω}(z) − G_{Ω}(w)| ≤ C_{1}|z − w|^{α}

for any pair of points z, w ∈ Ω(r_{0}), (ii) there exists a constant C_{2} such that

G_{Ω}(z) ≤ C_{2}δ_{Ω}(z)^{α}

for any point z ∈ Ω(r_{0}). In fact, we have C_{2} ≤ C_{1} ≤ 2^{1+α}C_{2} for the possible smallest constants C_{1} and C_{2} in the above.

Proof. For simplicity, we write G = G_{Ω}. By regularity of the boundary, G(z) → 0 as z → ζ ∈ ∂Ω. Hence, condition (ii) immediately follows from (i) with the same constant. Now we show the opposite direction. Assume condition (ii) and let z, w ∈ Ω(r_{0}). Case 1: 2|z − w| ≥ max{δ_{Ω}(z), δ_{Ω}(w)}.

In this case, we have

|G(z) − G(w)| ≤ G(z) + G(w) ≤ C_{2}(δ_{Ω}(z)^{α} + δ_{Ω}(w)^{α}) ≤ 2C_{2}(2|z − w|)^{α}.

Case 2: 2|z − w| < max{δ_{Ω}(z), δ_{Ω}(w)}.

We assume that 2|z − w| < δ_{Ω}(z) and take an arbitrary δ ∈ (0, δ_{Ω}(z)).

For any ζ ∈ 𝔻, we have the Poisson’s formula

G(z + δζ) = \frac{1}{2π} ∫_{0}^{2π} \frac{1 − |ζ|^{2}}{|e^{iθ} − ζ|^{2}} G(z + δe^{iθ})dθ.

Here note the elementary estimate |1 − (1 − |ζ|^{2})/|e^{iθ} − ζ|^{2}| ≤ 2|ζ|/(1 − |ζ|) ≤ 4|ζ| for |ζ| ≤ 1/2. Letting δ sufficiently near to δ(z), we can make the modulus of ζ := (w − z)/δ less than 1/2. Thus

|G(w) − G(z)| ≤ 4|ζ| · \frac{1}{2π} ∫_{0}^{2π} G(z + δe^{iθ})dθ = 4|ζ|G(z) = 4\frac{|w − z|}{δ}G(z). (6.1)

Letting δ → δ_{Ω}(z), we have

|G(w) − G(z)| ≤ 4\frac{|w − z|}{δ_{Ω}(z)}G(z) ≤ 4C_{2}|w − z|δ_{Ω}(z)^{α−1}

= 4C_{2}|w − z|^{α} (\frac{|w − z|}{δ_{Ω}(z)})^{1−α}

< 4C_{2}|w − z|^{α} (\frac{1}{2})^{1−α} = 2^{1+α}C_{2}|w − z|^{α}.

In any case, we have the first condition with C_{1} = 2^{1+α}C_{2}.

6.2. Hölder continuity of Green’s function. By the aid of the previous lemma, we can prove the following. Note that, in the case when ∞ ∈ Ω, the domain Ω satisfies the uniform LHMD property if and only if ∂Ω is uniformly perfect (see Theorem 4.1).

Theorem 6.2. Let Ω be a plane domain satisfying the uniform LHMD property with exponent α > 0. Then Green’s function of Ω is Hölder continuous with exponent α near the boundary.

Proof. As before, G denotes Green’s function of Ω with pole at ∞. By assumption, the local harmonic measure ω_{a,r,Ω} satisfies ω_{a,r,Ω}(z) ≤ C(|z − a|/r)^{α} for a ∈ ∂Ω, 0 < r ≤ r_{0}, where C and r_{0} are positive constants. We denote by K the supremum of G(z) in the set Ω(2r_{0}). Note that G can be regarded as the PWB solution H^{Ω(2r_{0})}φ for φ, where φ = 0 on ∂Ω and φ = G on ∂Ω(2r_{0}) ∩ Ω.

Let z ∈ Ω(r_{0}). Then we can take a point a ∈ ∂Ω with |z − a| = δ_{Ω}(z) = δ_{Ω(2r_{0})}(z). We now make use of the latter part of Theorem 2.2 to claim

G(z) ≤ CK (\frac{|z − a|}{r_{0}})^{α} = CKr_{0}^{−α}δ_{Ω}(z)^{α}.

Now Lemma 6.1 immediately yields the desired result.

6.3. Applications to uniformly perfect sets. As we have seen, an open set Ω has uniformly perfect boundary if and only if Ω satisfies global LHMD property (Theorem 4.1). Therefore, the next result, which was first stated in [8], immediately follows as a corollary.

Corollary 6.3. Let Ω ⊂ ℂ̂ be a domain whose boundary is uniformly perfect. Then Green’s function of Ω is Hölder continuous up to the boundary.

Using the quantitative results obtained before, we could have explicit estimates for the exponent of Hölder continuity of Green’s function in terms of constants characterizing uniform perfectness, however those would be, more or less, indirect.

We also note that the following result is obtained by combining with a theorem of Carleson [7], which asserts the removability of a compact set of Hausdorff dimension less than α for Hölder continuous harmonic functions with exponent α off the set.

Corollary 6.4. Let α be a number with 0 < α ≤ 1. If an open set Ω ⊂ ℂ satisfies the uniform LHMD property with exponent α, then Hausdorff dimension of its boundary is at least α.

In particular, a uniformly perfect set has positive Hausdorff dimension (cf. [8]). A similar result can be found in [41, Theorem 7.2]. If a compact set E contains a continuum, then, of course, E has Hausdorff dimension at least 1. Therefore, the above corollary actually makes sense only if ∂Ω is totally disconnected. We know abundant examples of totally disconnected, uniformly perfect sets such as limit sets in the complex dynamics as we noted in Introduction.

6.4. Historical remarks. The present research was originally motivated by the abovementioned fact that Green’s function on a domain with uniformly perfect boundary has a Hölder continuous extension to the boundary. We now explain a little bit more about the history.

In the book [8] of Carleson and Gamelin published in 1993, it is stated without proof that Green’s function of a domain with uniformly perfect boundary can be extended to the boundary in a Hölder continuous way. (In [9], an outline of the proof is presented in the case that the domain is a Fatou set, i.e., the complement of a Julia set.) They remarked that, from this fact, it follows that the Julia set of a rational function of degree at least two is of positive Hausdorff dimension as was stated above.

The first complete proof of this fact appeared in J. Lithner’s paper [26] in the literature. He showed that a compact set is uniformly perfect if and only if the set preserves the local Markov inequality. To prove that the local Markov inequality implies the global one, he used and proved the above fact (see also §8.1). His method consists of two steps; the first is to show that a uniformly perfect set necessarily contains a Cantor set with some uniform property, and the second is to show the complement of such a Cantor set carries Green’s function with Hölder continuity property.

Later, J. Siciak gave in [37] an explicit exponent of Hölder continuity of Green’s function in terms of condenser capacity density. He also noted in [37] the history of this result around 1994.

6.5. Applications to domains with uniform exterior conditions. We also have the following results from Theorems 5.2 and 5.5.

Corollary 6.5. If a plane domain Ω satisfies the uniform exterior circle condition, Green’s function of Ω is Lipschitz continuous near the boundary.

Corollary 6.6. If a plane domain Ω satisfies the uniform generalized exterior wedge condition with opening β, then Green’s function of Ω is Hölder continuous with exponent π/(2π − β) near the boundary.

In particular, any hyperbolic simply connected domain satisfies the generalized exterior wedge condition with opening 0, and thus its Green’s function is Hölder continuous with exponent 1/2, which is a classical result (see also Proposition 7.1 below).

6.6. Lipschitz spaces. Next, we consider the (global) Hölder continuity of PWB solutions for bounded, Hölder continuous boundary functions.

To make our results more comprehensive, it is desirable to employ the Banach spaces of Hölder continuous functions on subsets of ℂ. For a constant α ∈ (0, 1] and for a subset E of ℂ, we denote by Λ_{α}(E) the (real) Banach space consisting of all (real-valued) functions φ on E such that

∥φ∥_{Λ_{α}(E)} := sup_{x∈E} |φ(x)| + sup_{x,y∈E,x≠y} \frac{|φ(x) − φ(y)|}{|x − y|^{α}} < ∞.

Sometimes, Λ_{α}(E) is called the Lipschitz space (or the Hölder space) on E with exponent α. Note that Λ_{β}(E) is continuously embedded in Λ_{α}(E) if β > α. For convenience, though somewhat artificially, we define Λ_{α}(E) as Λ_{α}(E \{∞}) even for a subset E of ℂ̂ containing ∞.

6.7. Hölder continuity of PWB solutions. Using this notation, we can state one of our main results as in the following.

Theorem 6.7. Let Ω be a plane domain satisfying uniform LHMD property with exponent α. Then, for any γ with 0 < γ < α, there exists a constant B such that ∥H^{Ω}φ∥_{Λ_{γ}(Ω)} ≤ B∥φ∥_{Λ_{γ}(∂_{b}Ω)} for all φ ∈ Λ_{γ}(∂_{b}Ω).

Assume the uniform LHMD property with exponent α for Ω, namely, there are some constants 1 ≤ C < +∞ and 0 < r_{0} ≤ +∞ such that ω_{a,r,Ω}(z) ≤ C(|z − a|/r)^{α} holds for all 0 < r < r_{0} and z ∈ Ω ∩ B^{∘}(a, r). Let z ∈ Ω and take a ∈ ∂_{b}Ω such that |z − a| = δ_{Ω}(z). Let φ ∈ Λ_{γ}(∂_{b}Ω) for some γ < α. Then, by Theorem 2.2, we have

|H^{Ω}φ(z) − φ(a)| ≤2^{α}C (r_{0}^{−γ} sup_{ζ∈∂_{b}Ω} |φ(ζ) − φ(a)| + \frac{1}{α − γ} sup_{ξ,η∈∂_{b}Ω} \frac{|φ(ξ) − φ(η)|}{|ξ − η|^{γ}}) |z − a|^{γ}
≤2^{α}C (2r_{0}^{−γ} + 1/(α − γ)) ∥φ∥_{Λ_{γ}(∂_{b}Ω)}δ_{Ω}(z)^{γ}.

Therefore, we can get the expected conclusion by the following lemma.

Lemma 6.8. Let α ∈ (0, 1] and φ ∈ Λ_{α}(∂_{b}Ω). Suppose that a constant C ≥ 1 satisfies the inequality

|H^{Ω}φ(z) − φ(a)| ≤ C∥φ∥_{Λ_{α}(∂_{b}Ω)}δ_{Ω}(z)^{α}

for all φ ∈ Λ_{α}(∂_{b}Ω), z ∈ Ω and a ∈ ∂_{b}Ω such that δ_{Ω}(z) = |z − a|. Then ∥H^{Ω}φ∥_{Λ_{α}(Ω)} ≤ 16C∥φ∥_{Λ_{α}(∂_{b}Ω)} holds for all φ ∈ Λ_{α}(∂_{b}Ω).

Proof. We write u = H^{Ω}φ for a given φ ∈ Λ_{α}(∂_{b}Ω). Let z, w ∈ Ω.

Case 1: 2|z − w| ≥ max{δ_{Ω}(z), δ_{Ω}(w)}.

Take a and b in ∂_{b}Ω such that |z − a| = δ_{Ω}(z) and |w − b| = δ_{Ω}(w). Then we have

|u(z) − u(w)| ≤ |u(z) − φ(a)| + |φ(a) − φ(b)| + |u(w) − φ(b)|
≤ C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − a|^{α} + ∥φ∥_{Λ_{α}(∂_{b}Ω)}|a − b|^{α} + C∥φ∥_{Λ_{α}(∂_{b}Ω)}|w − b|^{α}
≤ (2^{1+α}C + 5^{α})∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α} ≤ 9C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α}

since |a − b| ≤ |a − z| + |z − w| + |w − b| ≤ 5|z − w|. Case 2: 2|z − w| < max{δ_{Ω}(z), δ_{Ω}(w)}.

We assume 2|z − w| < δ_{Ω}(z) and take a point a in ∂_{b}Ω such that δ_{Ω}(z) = |z − a|. Under the additional assumption that φ(a) = 0, we first show the following estimate:

|u(z) − u(w)| ≤ 8C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α}. (6.2)

We can write φ = φ^{+} − φ^{−}, where φ^{+} = (|φ| + φ)/2 and φ^{−} = (|φ| − φ)/2. Here note that ∥φ^{±}∥_{Λ_{α}(∂_{b}Ω)} ≤ ∥φ∥_{Λ_{α}(∂_{b}Ω)}. Set u^{+} = H^{Ω}φ^{+} and u^{−} = H^{Ω}φ^{−}. Then, in the same way as in the proof of (6.1), we can show

|u^{+}(z) − u^{+}(w)| ≤ \frac{4|z − w|}{δ_{Ω}(z)}u^{+}(z).

By assumption u^{+}(z) = u^{+}(z) − φ^{+}(a) ≤ C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − a|^{α}, we have

|u^{+}(z) − u^{+}(w)| ≤ 4C∥φ∥_{Λ_{α}(∂_{b}Ω)} (\frac{|z − w|}{δ_{Ω}(z)})^{1−α} |z − w|^{α} ≤ 4C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α}.

Similarly, we have |u^{−}(z) − u^{−}(w)| ≤ 4C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α}. Now we have estimate (6.2). In the case φ(a) ≠ 0, we replace φ by φ − φ(a), then have

|u(z) − u(w)| ≤ 8C∥φ − φ(a)∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α} ≤ 16C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α}.

Summing up the above, we have |u(z) − u(w)| ≤ 16C∥φ∥_{Λ_{α}(∂_{b}Ω)}|z − w|^{α} for any z and w in Ω, and thus the proof is complete.

Remark. By the above proof, we get information on the operator norm of H^{Ω} : Λ(∂_{b}Ω) → Λ(Ω), namely,

∥H^{Ω}φ∥_{Λ_{γ}(Ω)} ≤ 2^{5+α}C (2r_{0}^{−γ} + 1/(α − γ)) ∥φ∥_{Λ_{γ}(∂_{b}Ω)}.

6.8. Operator norm of harmonic extension. Combining Theorems 5.2, 5.5 and the remark just above, we obtain the following results.

Corollary 6.9. If a plane domain Ω satisfies the uniform exterior circle condition with radius ρ, for any α ∈ (0, 1), the harmonic extension operator H^{Ω} : Λ_{α}(∂_{b}Ω) → Λ_{α}(Ω) is bounded and its operator norm satisfies

∥H^{Ω}∥_{Λ_{α}} ≤ C (ρ^{−α} + \frac{1}{1 − α}) , (6.3)

where C is an absolute constant.

Corollary 6.10. If a plane domain Ω satisfies the uniform generalized exterior wedge condition with opening β, for any α < π/(2π − β), the harmonic extension operator H^{Ω} : Λ_{α}(∂_{b}Ω) → Λ_{α}(Ω) is bounded.

In the proof of Theorem 6.7, for a given z ∈ Ω, we needed only to consider a boundary point a with δ(z) = |z − a|. In this case, we have p_{Ω}(z, a) = |z − a| = δ(z). Hence, from Theorem 5.3, we also have the following corollary.

Corollary 6.11. Let Ω be a plane domain whose boundary is of class C^{2} and has curvature not less than −1/ρ, where ρ is a positive constant. Then the harmonic extension operator H^{Ω} : Λ_{α}(∂_{b}Ω) → Λ_{α}(Ω) satisfies the same norm estimate as (6.3) for an absolute constant C.

6.9. Examples. We cannot expect that the above results would be still valid in general when γ = α, as the following simple examples show.

Example 6.1. We take the unit disk 𝔻 as Ω. Consider the function φ(ζ) = |Im ζ| on ∂𝔻, which is, of course, Lipschitz continuous on ∂𝔻. But the harmonic extension u of φ is not Lipschitz continuous in 𝔻 because u(r), where −1 < r < 1, can be expressed by

u(r) = \frac{1}{2π} ∫_{0}^{2π} \frac{1 − r^{2}}{1 − 2r cos t + r^{2}} | sin t|dt = \frac{1 − r^{2}}{πr} log \frac{1 + r}{1 − r}.

Therefore, the image of Λ_{1}(∂𝔻) under the harmonic extension operator H^{𝔻} to 𝔻 is not contained in Λ_{1}(𝔻). Note that a similar example was constructed by Hinkkanen (Example (5.11) of [19]).

By transformation under the map h(z) = (1 − z)/(1 + z), we obtain the right half plane Ω = ℍ_{+} = {z; Re z > 0} and the boundary function ψ(iy) = 2|y|/(1 + y^{2}), whose harmonic extension v has the expression

v(x) = \frac{2x}{π(x^{2} − 1)} log x

on the positive real axis ℝ_{+}.

In order to show that Corollary 6.10 is sharp, we bend the above example.

Example 6.2. For 1/2 ≤ α ≤ 1, we consider the domain ℍ_{+}^{α} = {z ∈ ℂ \ {0}; | arg z| < π/2α}. This satisfies the uniform (generalized) exterior wedge condition with opening β = 2π − π/α. By Theorem 5.5, ℍ_{+}^{α} has the global LHMD property with exponent α = π/(2π − β). Hence Corollary 6.10 ensures boundedness of the harmonic extension operator H from Λ_{γ}(∂_{b}ℍ_{+}^{α}) into Λ_{γ}(ℍ_{+}^{α}) for γ < α. However, the corresponding statement for γ = α no longer holds. Indeed, the function θ(re^{±πi/2α}) = ψ(±ir^{α}) = 2r^{α}/(1 + r^{2α}) belongs to Λ_{α}(∂_{b}ℍ_{+}^{α}) while its harmonic extension w(z) = v(z^{α}) has the expression

w(x) = \frac{2αx^{α}}{π(x^{2α} − 1)} log x

on the positive real axis ℝ_{+}, where ψ and v are the functions defined in the previous example, in particular, w ∉ Λ_{α}(ℍ_{+}^{α}).

To obtain a bounded domain Ω with the same property when 1/2 < α < 1, we have only to transform ℍ_{+}^{α} by h(z) = (1 − z)/(1 + z). The resulting domain is

h^{−1}(ℍ_{+}^{α}) = \{z ∈ ℂ; |z + i cot \frac{π}{2α}| < \frac{1}{sin \frac{π}{2α}} or |z − i cot \frac{π}{2α}| < \frac{1}{sin \frac{π}{2α}} \} .

We note that we can modify this example to obtain a bounded domain Ω with global LHMD property with exponent 1/2 for which H^{Ω} does not preserve Hölder spaces with exponent 1/2. However, when 0 < α < 1/2, the author does not know if we could get a (bounded) domain satisfying global LHMD property with exponent α for which the harmonic extension operator does not preserve Hölder spaces with exponent α.

Remark. Quite recently, Aikawa [2] proved the remarkable result that there is no bounded domain Ω in ℝ^{n} (n ≥ 2) for which H^{Ω}(Λ_{1}(∂Ω)) ⊂ Λ_{1}(Ω) holds. Moreover, he established the converse of Theorem 6.7, in other words, for a bounded domain Ω with regular boundary, if the harmonic extension operator H^{Ω} preserves Λ_{α} for some α > 0, then Ω enjoys the global LHMD property with exponent α. In particular, a bounded domain Ω with regular boundary has uniformly perfect boundary if and only if H^{Ω}(Λ_{α}(∂Ω)) ⊂ Λ_{α}(Ω) for some α > 0.

7. Application to conformal mappings

7.1. Boundary behaviour of a conformal mapping and Green’s function. In this section, we present applications of Theorem 6.2 to the theory of conformal mappings. We consider a finitely connected bounded domain Ω in ℂ without degenerate boundary components even if the boundary is bounded. As is well known, the domain Ω is can be mapped by a conformal mapping g onto a plane domain, say Ω_{0}, bounded by finitely many closed analytic curves. Let f = g^{−1} : Ω_{0} → Ω. Here and hereafter an analytic curve will mean a simple closed curve of class C^{ω} in the complex plane ℂ. (Furthermore, by Koebe’s theorem, we can choose Ω_{0} so that each component of its boundary is a circle.) Take a point a ∈ Ω and set a_{0} = g(a). We denote by G and G_{0} the Green’s functions of Ω and Ω_{0} with pole at a and a_{0}, respectively. Hence, G(f(z)) = G_{0}(z) for each z ∈ Ω_{0}.

By conformal invariance of the hyperbolic metric, we note the identity λ_{Ω}(f(z))|f′(z)| = λ_{Ω_{0}}(z). Since ∂Ω is uniformly perfect, by Theorem B,

c ≤ δ_{Ω}(w)λ_{Ω}(w) ≤ 1 (7.1)

holds for some positive constant c.

Let α be a constant with 0 < α ≤ 1. By Lemma 6.1, G is Hölder continuous of exponent α near the boundary if and only if G(w) ≤ Cδ_{Ω}(w)^{α} holds near the boundary for some constant C. By (7.1), this property is equivalent to the condition

G_{0}(z) = G(f(z)) ≤ const.λ_{Ω}(f(z))^{−α} = const.(|f′(z)|/λ_{Ω_{0}(z)})^{α}

when z is sufficiently close to ∂Ω_{0}. By using Schwarz’ reflection principle, we can see G_{0}(z) ≍ δ_{Ω_{0}}(z) ≍ 1/λ_{Ω_{0}}(z) near the boundary of Ω_{0}. Now we can easily verify the following proposition.

Proposition 7.1. Let Ω be a finitely connected bounded domain in ℂ without punctures. Let α ∈ (0, 1] be a constant and f : Ω_{0} → Ω be a conformal mapping, where Ω_{0} is a plane domain bounded by finitely many analytic curves. Then, Green’s function of Ω is Hölder continuous of exponent α near the boundary if and only if there exists a positive constant m such that

\frac{1}{|f′(z)|} ≤ mλ_{Ω_{0}}(z)^{\frac{1}{α} −1} (7.2)

for z ∈ Ω_{0}.

If Ω is simply connected and if Ω_{0} is the unit disk, the inequality (7.2) with α = 1/2 immediately follows from the Koebe distortion theorem for univalent functions. This, of course, agrees with the statement of Corollary 6.6.

Now Corollary 6.6 yields the next result, which is a natural generalization of Theorem I in [28] shown by M. Masumoto.

Corollary 7.2. Let f : Ω_{0} → Ω be as above. Suppose that Ω satisfies the uniform generalized exterior wedge condition with opening β ∈ [0, π]. Then there exists a positive constant m such that

\frac{1}{|f′(z)|} ≤ mλ_{Ω_{0}}(z)^{1−β/π}

for z ∈ Ω_{0}.

Note that an arbitrary, finitely connected, bounded domain without degenerate boundary components satisfies the generalized wedge condition with opening 0.

7.2. Non-integrability of subharmonic functions. By virtue of the above corollary, the same argument as in [28] yields the following result.

Theorem 7.3. Let Ω be a finitely connected bounded domain in ℂ without punctures. Suppose that Green’s function of Ω is Hölder continuous of exponent α ∈ (0, 1] near the boundary. We set γ(p, α) = 2 − α min{1, p} for 0 < p < ∞. If a nonnegative subharmonic function s on Ω satisfies

∫∫ _{Ω} δ_{Ω}(z)^{−γ(p,α)}s(z)^{p}dxdy < +∞, z = x + iy

for some 0 < p < ∞, then s must vanish identically.

In the case when the boundary of Ω is of class C^{1,1}, N. Suzuki [42] proved the above result with α = 1. Actually, our general statement can be deduced from Suzuki’s special one as is described in [28]. For convenience, we now reproduce it.

Proof. Let γ = γ(p, α) and γ_{0} = γ(p, 1). Note γ ≤ 2 and that s_{0} := s∘f is subharmonic in the domain Ω_{0}. By (7.1), we may suppose that s satisfies ∫∫_{Ω} λ_{Ω}(w)^{γ}s(w)^{p}dudv < +∞. Noting estimate (7.2), by a change of variables, we have

∫∫ _{Ω} λ_{Ω}(w)^{γ}s(w)^{p}dudv = ∫∫ _{Ω_{0}} λ_{Ω_{0}}(z)^{γ}|f′(z)|^{2−γ}s_{0}(z)^{p}dxdy

≥ \frac{1}{m} ∫∫ _{Ω_{0}} λ_{Ω_{0}}(z)^{γ_{0}}s_{0}(z)^{p}dxdy.

Hence, by Suzuki’s result, we have s_{0} = 0.

Corollary 7.4. If a finitely connected bounded domain Ω satisfies the uniform generalized exterior wedge condition with opening β ∈ [0, π], then there does not exist a non-negative subharmonic function s ≠ 0 on Ω with

∫∫ _{Ω} δ_{Ω}(z)^{−γ_{p}}s(z)^{p}dxdy < +∞, z = x + iy

for some 0 < p < ∞, where γ_{p} = 2 − min{p, 1}/(2 − β/π).

7.3. Integrability of the derivatives of conformal mappings. We now give another application of estimate (7.2). Consider the inverse mapping g : Ω → Ω_{0} of f. It is trivial that ∫∫_{Ω} |g′(w)|^{2}dudv = Area (Ω_{0}) < +∞. In our case, we can say more.

Theorem 7.5. Let Ω be a finitely connected bounded domain without punctures of which Green’s function is Hölder continuous with exponent α. Suppose that g is a conformal mapping from Ω onto a domain Ω_{0} bounded by finitely many analytic curves. Then we have

∫∫ _{Ω} |g′(z)|^{p}dxdy < +∞

for any 0 < p < 2 + α/(1 − α). In particular, for a domain Ω satisfying the generalized wedge condition with opening β, we have the above result for any 0 < p < 2 + π/(π − β).

Proof. We will give a proof for the case that Ω is simply connected and that Ω_{0} is the unit disk 𝔻. The general case can be treated in the quite same way as follows.

We may assume 2 < p < 2 + α/(1 − α). By (7.2), we have

∫∫ _{Ω} |g′(w)|^{p}dudv = ∫∫ _{𝔻} |f′(z)|^{2−p}dxdy ≤ M ∫∫ _{𝔻}(1 − |z|)^{−(p−2)(1−α)/α}dxdy
≤ 2πM ∫_{0}^{1}(1 − r)^{−(p−2)(1−α)/α}dr,

where M is a constant. The last term is finite if (p − 2)(1 − α)/α > 1.

In particular, if Ω is bounded and simply connected and if Ω_{0} is the unit disk, the above yields ∫∫_{Ω} |g′(w)|^{p}dudv < +∞ for any 0 < p < 2 + α/(1 − α). Note again that α ≥ 1/2 always holds, and hence 2 + α/(1 − α) ≥ 3.

We should remark that the Brennan conjecture asserts that ∫∫_{Ω} |g′(w)|^{p}dudv < +∞ for any simply connected domain Ω and for any 0 < p < 4, but this is unsolved yet so far. The best result up to now is p < 3.39, which is due to Ch. Pommerenke (see [34, Chapter 8]). Therefore, our result above must not be sharp.

Finally, we mention a couple of results involving the exterior wedge condition. F. D. Lesley [25] proved that for a bounded Jordan domain Ω with exterior wedge condition of opening β > 0 the inverse of the Riemann mapping function g = f^{−1} : Ω → 𝔻 is Hölder continuous with exponent 1/(2 − β/π). Under the same condition, C. H. Fitzgerald and Lesley [13] proved that 1/f′ ∈ H^{p} for 0 < p < 1/(1 − β/π), where H^{p} denotes the Hardy space on the unit disk with exponent p. Our results above are closely related to their results, however, there seem to be no immediate implications between them.

8. Counterexample

8.1. Application to Markov inequalities. We construct here a simple counterexample showing that Hölder continuity property of Green’s function does not necessarily imply uniform perfectness of the boundary. Actually, our domain here carries Lipschitz continuous Green’s functions, whereas its boundary is not uniformly perfect. This answers negatively one of the questions raised in [37]. J. Lithner [26] has constructed a certain kind of Cantor set preserving the local Markov inequality, but not preserving the global Markov inequality. Lithner [26] also proved that the global Hölder continuity of Green’s function implies the global Markov inequality, as well as that uniform perfectness is equivalent to the preservation of local Markov inequality. Therefore, our example gives another example similar to Lithner’s one (also note that our example has no degenerate boundary components).

8.2. Construction. Let a sequence r_{n} of positive numbers satisfying r_{n} < 1/8 be given. Let ε_{n} and M_{n} (n = 1, 2, …) be sequences of positive numbers monotonically decreasing to 0, which we will specify later. We take ε_{1} < 1/2. Set

Ω̂_{0} ={z ∈ ℂ; |Im z| < 1/2, Re z > −1}

\ ⋃_{n=1}^{∞}(B(2n − 1 + (1 + ε_{n})i/2, 1/2) ∪ B(2n − 1 − (1 + ε_{n})i/2, 1/2)).

Note that Ω̂_{0} is simply connected and satisfies the exterior circle condition with constant ρ = 1/2. We now consider the domain Ω̂ := Ω̂_{0} \ ⋃_{n=1}^{∞} B(2n, r_{n}) and its inversion Ω := {z ∈ ℂ̂; 1/z ∈ Ω̂}. Let G be the Green’s function of Ω with pole at infinity and Ĝ be the Green’s function of Ω̂ with pole at the origin, and hence G(1/z) = Ĝ(z). If we denote by Ĝ_{0} the Green’s function of Ω̂_{0} with pole at the origin, then we have Ĝ < Ĝ_{0} on Ω̂ by the minimum principle. Note that the majorant Ĝ_{0} does not depend on the sequence r_{n}.

We begin with the comparison between δ := δ_{Ω} and δ̂:= δΩ̂. Set D_{n} := {z ∈ Ω̂; 2n−1 < Re z < 2n + 1} and D̃_{n} := {z ∈ Ω̂_{0}; 2n − 1 < Re z < 2n + 1}.

Lemma 8.1. For z ∈ D_{n} (n ≥ 0), we have δ̂(z) ≤ 4(n + 1)^{2}δ(1/z).

Proof. Let a ∈ ∂Ω̂ such that δ(1/z) = |1/z − 1/a| = |z − a|/|az|. Then the point a is contained in the boundary of a disk in Ω̂ containing z. Thus |a| ≤ 2n + 1 + ε(1 + ε/2) ≤ 2n + 2. Now we have δ(1/z) ≥ |z − a|/4(n + 1)^{2} ≥ δ̂(z)/4(n + 1)^{2}.

It is easy to see that, for a given sequence M_{n} of positive numbers, there exists a sequence ε_{n} such that Ĝ_{0} ≤ M_{n} on D̃_{n} (n = 1, 2, …). The maximum principle implies that Ĝ ≤ M_{n} on D_{n} for each n = 1, 2, …. By the proof of Theorem 6.2, we can show Ĝ(z) ≤ CM_{n}δ̂_{0}(z) on D_{n}, where δ̂_{0} = δΩ̂_{0} and C is an absolute constant not less than one. If z ∈ D_{n} satisfies δ̂_{0}(z) = δ̂(z), then we have Ĝ(z) ≤ CM_{n}δ̂(z). If not, it is clear that the point z satisfies |z − 2n| < 1/2. We set A_{n} := {z ∈ ℂ; r_{n} < |z − 2n| < 1/2} and denote by ω_{n} the harmonic measure of {|z − 2n| = 1/2} relative to A_{n} for each n = 1, 2, ….

Explicitly, we can write ω_{n}(z) = (log |z − 2n|/r_{n})/μ_{n}, where μ_{n} denotes the modulus of A_{n}, i.e., μ_{n} = log 1/2r_{n}. Then we see that Ĝ ≤ M_{n}ω_{n} on A_{n}. In other words,

Ĝ(z) ≤ \frac{M_{n}}{μ_{n}} log \frac{|z − 2n|}{r_{n}} ≤ \frac{M_{n}}{μ_{n}} \frac{|z − 2n| − r_{n}}{r_{n}} = \frac{M_{n}}{μ_{n}r_{n}} δ̂(z).

Now we take

M_{n} = \frac{μ_{n}r_{n}}{4(n + 1)^{2}} = \frac{r_{n} log 1/2r_{n}}{4(n + 1)^{2}} .

Then we have Ĝ(z) ≤ Cδ̂(z)/4(n + 1)^{2} for z ∈ D_{n}. This estimate can be made still valid for n = 0 and z ∈ D_{0} \ B(0, 1/4) by replacing the constant C by another one if necessary. Now, by Lemma 8.1, we have G(z) ≤ Cδ(z) for z ∈ Ω with |z| ≤ 4, which implies the Lipschitz continuity of G near the boundary (cf. Lemma 6.1).

Choosing the sequence r_{n} so that r_{n} → 0 as n → ∞, the moduli μ_{n} of essential annuli A_{n} of Ω̂ tend to infinity as n → ∞, therefore Ω̂, and hence Ω does not have uniformly perfect boundary (cf. [41]).

9. Appendix

In this appendix, we will present a proof of Proposition 5.4. To this end, first we give preparatory results which are true also for regular plane curves with the same curvature bound.

Let κ be a continuous function from an interval I with 0 ∈ I into ℝ. Denote by Δ(ρ) the open disk of radius ρ and centered at −iρ. We set θ(t) = ∫_{0}^{t} κ(s)ds and consider the regular curve ξ : I → ℂ given by ξ(t) = ∫_{0}^{t} e^{iθ(s)}ds. We write ξ(t) = X(t) + iY (t), in other words,

(9.1) X(t) = ∫_{0}^{t} cos θ(s)ds and Y (t) = ∫_{0}^{t} sin θ(s)ds.

Of course, the curve ξ is not necessarily simple, but the following lemmas can be shown. (In the case when |κ| ≤ 1/ρ, we can deduce stronger global properties for the curve. See, for example, [20] and the references cited therein.)

Lemma 9.1. Suppose that θ(t_{0}) = 0, θ(t_{1}) = −π/2, κ(t) = θ′(t) ≥ −1/ρ and 0 ≥ θ(t) ≥ −π/2 for t_{0} ≤ t ≤ t_{1}, where ρ is a positive constant. Then

(9.2) X(t) − X(t_{0}) ≥ ρ sin \frac{t − t_{0}}{ρ} and Y (t) − Y (t_{0}) ≥ ρ (cos \frac{t − t_{0}}{ρ} − 1)

for t_{0} ≤ t ≤ t_{1}, in particular, the curve ξ(t) − ξ(t_{0}) is injective and does not intersect the disk Δ(ρ) on the interval [t_{0}, t_{1}]. Moreover, the following hold:

X(t_{1}) − X(t_{0}) ≥ ρ and Y (t_{1}) − Y (t_{0}) ≤ −ρ. (9.3)

Proof. Since −π/2 ≤ θ(t_{0} + s) ≤ −s/ρ, we see sin θ(t_{0} + s) ≥ − sin s/ρ and cos θ(t_{0} + s) ≥ 1 − cos s/ρ. Now inequalities (9.2) follow from the expression (9.1). In particular, letting t = t_{1}, we have the first inequality in (9.3). To prove the second inequality in (9.3), we need more analysis.

We define the monotone decreasing continuous function θ̂ by

θ̂(t) = min_{t_{0}≤s≤t} θ(s).

Then θ̂(t) ≤ θ(t). Noting that −π/2 − θ(t) = ∫_{t}^{t_{1}} θ′(s)ds ≥ −(t_{1} − t)/ρ, we have t_{1} − t ≥ ρ(θ(t) + π/2) ≥ ρ(θ̂(t) + π/2) for t_{0} ≤ t ≤ t_{1}.

Since θ = θ̂ on the support E of the probability measure −d sin θ̂(t) on [t_{0}, t_{1}] and since sin θ(t) ≤ 0 on [t_{0}, t_{1}], by integrations by parts, we have

Y (t_{1}) − Y (t_{0}) = ∫_{t_{0}}^{t_{1}} sin θ(t)dt ≤ ∫_{E} sin θ̂(t)dt
= − ∫_{E}(t − t_{1})d sin θ̂(t) = − ∫_{t_{0}}^{t_{1}}(t − t_{1})d sin θ̂(t)

≤ ∫_{t_{0}}^{t_{1}} ρ(θ̂(t) + π/2)d sin θ̂(t) = ρ ∫_{0}^{−π/2}(θ + π/2)d sin θ = −ρ.

Lemma 9.2. Suppose that θ(t_{0}) = 0, θ′(t) ≥ −1/ρ and −π/2 ≤ θ(t) ≤ π/2 for t_{0} ≤ t ≤ t_{1}. Then the curve ξ(t) − ξ(t_{0}) is injective and does not intersect the disk Δ(ρ) on the interval [t_{0}, t_{1}].

Proof. First we observe, by assumption, X(t) is non-decreasing in [t_{0}, t_{1}]. By the approximation argument, we may assume that the function θ has at most finitely many zeros in [t_{0}, t_{1}]. We divide the interval [t_{0}, t_{1}] into subintervals I_{1}, …, I_{N} with I_{j} = [a_{j}, b_{j}] such that a_{1} = t_{0}, b_{j} = a_{j−1} and b_{N} = t_{1}, |θ| > 0 on (a_{j}, b_{j}) and that θ(a_{j}) = 0 for j = 1, …, N. We assume that the assertion in the lemma is true for I_{1} ∪ ⋯ ∪ I_{n−1}. In the case when θ > 0 on I_{n}, Y (t) is monotone increasing in this interval, therefore the assertion is true also for I_{1} ∪ ⋯ ∪ I_{n}. In the case when θ < 0 on I_{n}, we can use Lemma 9.1 to conclude truth of the assertion for this case. Hence, by induction, we have shown the assertion on the whole interval [t_{0}, t_{1}].

Lemma 9.3. Suppose that θ(t_{0}) = 0, θ(t_{1}) = −π, θ′(t) ≥ −1/ρ and 0 ≥ θ(t) ≥ −π for t_{0} ≤ t ≤ t_{1}. Then Y (t) is non-increasing, in particular, ξ is injective on this interval [t_{0}, t_{1}]. Furthermore the curve ξ(t) − ξ(t_{0}) does not intersect the disk Δ(ρ) and the Euclidean diameter of ξ([t_{0}, t_{1}]) is at least 2ρ.

Proof. The first assertion easily follows from Y′(t) = sin θ(t) ≤ 0. Next let s_{0} = min{s > 0; θ(s) = −π/2}. By Lemma 9.1, ξ([t_{0}, s_{0}]) does not intersect Δ(ρ) and the endpoint X(s_{0}) + iY (s_{0}) satisfies X(s_{0}) ≥ ρ and Y (s_{0}) ≤ −ρ. Now Lemma 9.2 is applicable for the curve iξ on the interval [s_{0}, t_{1}] to conclude that ξ(t) − ξ(t_{0}) does not intersect the disk Δ(ρ). Let s_{1} = max{s > 0; θ(s) = −π/2}. Applying Lemma 9.1 to the curve ξ([s_{1}, t_{1}]), we obtain Y (t_{1}) − Y (s_{0}) ≤ Y (t_{1}) − Y (s_{1}) ≤ −ρ. Therefore, the last assertion now follows from this and Y (s_{0}) ≤ −ρ.

Of course, to obtain the statement of Proposition 5.4, we must make essential use of simpleness of the curve ξ. Before the proof, observe that the mirror image ξ̃, which will be defined here by ξ̃(t) = −\overline{ξ(−t)}, has the same curvature bound because κ̃(t) = κ(−t).

We consider the situation stated in the proposition. Let L be the length of the curve ξ. Since ξ is simple, the rotation index ∫_{0}^{L} κ(t)dt/2π is ±1 by the theorem of turning tangents (cf. [10]). The signature depends only on the orientation of the curve. Note that this notion can easily be defined and the above is still true in the case that the curve has finitely many corners. See [34] for more detailed account.

Without loss of generality, we may assume that a = ξ(0) = 0 and that ξ′(0) = 1, and hence Δ = Δ(ρ) = {z; |z + iρ| < ρ}. Then, by the function θ(t) = ∫_{0}^{t} κ(s)ds, the unit velocity vector ξ′(t) can be represented as exp(iθ(t)).

Now suppose that the curve ξ intersects Δ. Set t_{∗} = inf{t > 0; ξ(t) ∈ Δ} and θ_{∗} = θ(t_{∗}). We can define φ(t) = arg(ξ(t)+iρ)−π/2 as a continuous function on [0, t_{∗}] with φ(0) = 0. We set φ_{∗} = φ(t_{∗}). From Lemma 9.2, |φ_{∗}| > 0 follows. Since ξ is simple, we note that |φ_{∗}| < 2π. If |φ_{∗}| ≥ π, we can quite easily show that diam ξ([t_{0}, t_{∗}]) ≥ 2ρ. Hence, we assume |φ_{∗}| < π in the sequel. Case 1: −π < φ_{∗} < 0.

We now consider the simple closed curve ξ_{∗} which consists of the subarc ξ|_{[0,t_{∗}]} and the two segments [iρ(e^{iφ_{∗}} − 1), −iρ] and [−iρ, 0]. Let ω be the turning tangent of ξ_{∗} at the point ξ(t_{∗}), then ω = φ_{∗} − π/2 − θ_{∗} + 2nπ for some integer n and |ω| ≤ π/2.

Then it is clear that the rotation index (θ_{∗} + ω + (−π − φ_{∗}) − π/2)/2π equals −1. This forces n = 0, and hence |φ_{∗} − θ_{∗} − π/2| ≤ π/2, i.e.,

φ_{∗} − π ≤ θ_{∗} ≤ φ_{∗} < 0. (9.4)

Next we consider the case when

sup_{0≤t<t′≤t_{∗}} |θ(t) − θ(t′)| ≥ π. (9.5)

In this case, there exist t_{0} and t_{1} with t_{0} < t_{1} in [0, t_{∗}] such that θ(t_{1}) − θ(t_{0}) = ±π and θ(t) lies between θ(t_{0}) and θ(t_{1}) for all t_{0} ≤ t ≤ t_{1}. If θ(t_{1}) − θ(t_{0}) = −π, then, essentially, Lemma 9.3 can be applied to obtain diam ξ([t_{0}, t_{1}]) ≥ 2ρ, and thus the expected assertion follows. Otherwise, we have only to consider the mirror image ξ̃ to deduce the same assertion.

Finally, suppose that (9.5) does not hold. We set θ_{max} = max_{0≤t≤t_{∗}} θ(t), t_{0} := min{t ∈ [0, t_{∗}]; θ(t) = θ_{max}} and Δ′ := {ξ′(t_{0})z + ξ(t_{0}); z ∈ Δ}.

By Lemma 9.3, the curve ξ([t_{0}, t_{∗}]) does not intersect the disk Δ′. By applying Lemma 9.3 to the mirror image ξ̃, we can show that the curve ξ([0, t_{0}]) does not intersect the disk Δ′, too.

Now the curve ξ([0, t_{∗}]) ∪ {e^{iθ}; 0 ≤ t ≤ t_{∗}} should enclose the domain D := (Δ′)^{∘} \ Δ, and thus diam ξ([0, t_{∗}]) ≥ diam D ≥ 2ρ. Case 2: 0 < φ_{∗} < π.

Suppose that diam ξ([0, t_{∗}]) < 2ρ. Let D be the domain bounded by this subarc ξ([0, t_{∗}]) and the segment [0, ξ(t_{∗})]. Then we have diam D < 2ρ.

We consider the mirror image ξ̃(−t) = −\overline{ξ(t)} of ξ. Then the subarc ξ̃([0, t̃_{∗}]), where t̃_{∗} is the parameter value determined for ξ̃ in the similar way to t_{∗} above, would be contained in the mirror image of D, and hence diam ξ̃([0, t̃_{∗}]) ≤ diam D < 2ρ, which is impossible by Case 1.

Now the proof of Proposition 5.4 is complete.

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Department of Mathematics, Kyoto University, 606-8502 Kyoto, Japan

Current address: Department of Mathematics, University of Helsinki, P.O.Box 4 (Yliopistonkatu 5), FIN-00014, Helsinki, Finland

  E-mail address: sugawa@kusm.kyoto-u.ac.jp