Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 143–172

GENERALIZED MEAN POROSITY AND DIMENSION

Tomi Nieminen

University of Jyväskylä, Department of Mathematics and Statistics P.O. Box 35, FI-40014 Jyväskylä, Finland; tominiem@maths.jyu.fi

Abstract. We define the class of weakly mean porous sets and prove a sharp dimension estimate for the sets in this class. By using this geometric tool, we establish an essentially sharp dimension bound for the boundaries of generalized Hölder domains and John domains.

1. Introduction

In this paper we consider the following problem. Suppose that we are given the growth condition

(1)  k_{Ω}(x_{0}, x) ≤ ϕ(\frac{d(x, ∂Ω)}{d(x_{0}, ∂Ω)}) + C_{0}

on the quasihyperbolic metric k_{Ω} of a domain Ω, where ϕ is a decreasing function and x_{0} is a fixed point in Ω. Under which conditions on the function ϕ can we prove a dimension estimate for the boundary ∂Ω, and what is the sharp dimension estimate in this case?

Let us comment on the history of this problem. Recall that a domain Ω satisfying condition (1) with the function ϕ(t) = C log(1/t) is called a Hölder domain (see e.g. [SS1]). It is well known that for a Hölder domain Ω ⊂ R^{n} we have the estimate dim_{H}(∂Ω) < n. This was proven by Smith and Stegenga [SS2] using ideas of Jones and Makarov [JM]. They established this result by applying Marcinkiewicz integrals. Later Koskela and Rohde [KR] proved a sharp extension of this result using a different technique. They introduced the concept of mean porosity and, as an application of this concept, they proved the sharp dimension estimate for the boundary of a Hölder domain. In this paper, we define a generalized version of mean porosity and, by applying this concept, we will prove an essentially sharp dimension estimate for the boundary of a domain Ω satisfying condition (1) with some decreasing function ϕ. Note that since the Hausdorff dimension of a Hölder domain can be arbitrarily close to n, one cannot hope for an estimate on the usual Hausdorff dimension with weaker assumptions. Instead we will use Carathéodory’s construction of measure and dimension, which allows us to measure the size of a set with much finer scale. For example, if a domain Ω ⊂ R^{n} satisfies condition (1) with ϕ(t) = (log(1/t))^{s}, then we will obtain a dimension bound when s ≤ n/(n − 1), whereas the boundary can have positive volume when s > n/(n − 1). In particular, for s = n/(n − 1), we prove that H^{h}(∂Ω) = 0 for the gauge function h(t) = t^{n}(log(1/t))^{C}.

2000 Mathematics Subject Classification: Primary 28A75; Secondary 30C65, 30F45.

The research was supported by Vilho, Yrjö and Kalle Väisälä foundation. This paper is a part of my PhD thesis written under the supervision of Professor Pekka Koskela. I wish to thank him for all the advisement and support. I also wish to thank Toni Heikkinen and Ignacio Uriarte-Tuero for useful discussions and Stanislav Hencl and the referee for comments on the manuscript.

Notice that the geometric problem introduced above can be considered also from the viewpoint of uniform continuity of quasiconformal mappings. Indeed, if f: B^{n} → R^{n} is a uniformly continuous quasiconformal mapping defined in the unit ball with a modulus of continuity ψ, then the image domain f(B^{n}) satisfies condition (1) with a corresponding function ϕ (see Section 5). For conformal mappings in the plane, the sharp condition for the function ψ implying m_{2}(∂f(B^{2})) = 0 is already known by [JM]. We will prove an extended result for quasiconformal mappings in R^{n} with n ≥ 2 and, moreover, we will prove an essentially sharp dimension estimate for ∂f(B^{n}).

Another question, related to our main problem, concerns John domains. It is well known that the Hausdorff dimension of the boundary of a usual c-John domain is strictly smaller than n, see [T], [MV], [KR]. But what can be said about the size of the boundary of a φ-John domain (see Section 6 for definition) with some function φ that is not linear? We will give a precise answer to this question in Chapter 6.

We obtain the results above by establishing a sharp dimension bound for sets satisfying a certain porosity condition. Roughly speaking, we require that, if we consider dyadic annuli A_{k}(x), k = 1, 2, …, centered at some point x ∈ E, then at least half of the annuli contain λ “holes” of size α. Here λ and α are some functions depending on the scale k. Moreover, we require that these cubes or “holes” can be picked for each point x from a single disjoint collection of cubes in the complement of E that does not depend on the point x. Thus our porosity condition is not strictly pointwise (as porosity conditions are in general). Also note that our porosity condition does not permit scaling. Nevertheless, our definition of generalized mean porosity works well from the viewpoint of our applications.

The paper is organized as follows. After establishing some notation and definitions in Section 2, we introduce the porosity condition in Section 3 and prove also the basic dimension estimate. Section 4 contains an application of generalized mean porosity to the domains satisfying a quasihyperbolic growth condition. In Section 5 we prove a corresponding result for the boundaries of image domains under uniformly continuous quasiconformal mappings. We discuss the properties of φ-John domains in Section 6 and, finally, in Section 7 we construct examples of sets showing the sharpness of the dimension estimates proven in this paper.

2. Notation and definitions

Throughout this paper we denote by R^{n}, n ≥ 1, the euclidean space of dimension n. The Lebesque measure of a set E ⊂ R^{n} is denoted by |E|, although we sometimes write m_{n}(E) to emphasize the dimension n. We denote the euclidean distance between two points x, y ∈ R^{n} by d(x, y). We define a neighborhood of E by E + r := {z ∈ R^{n} : d(z, E) < r}, where r > 0 and d(z, E) = inf{d(z, x) : x ∈ E}.

We set Z^{+} := {1, 2, 3, …}. For x ∈ R^{n} we denote by A_{k}(x) the set

A_{k}(x) = {y ∈ R^{n} : 2^{−k} < |x − y| < 2^{−k+1}},

where k ∈ Z^{+} . We denote by ♯I the number of elements in the set I.

For a cube Q ⊂ R^{n} we denote by l(Q) the edge length and by d(Q) the diameter of Q. The radius of a ball B ⊂ R^{n} is denoted by r(B). We denote by pB, p > 0, a ball with the same center as B but with radius pr(B). We write B^{n} ⊂ R^{n} for the unit ball centered at the origin with radius 1. We denote the unit sphere by S^{n−1} .

Let γ ⊂ R^{n} be an injective curve and let x, y ∈ γ. We denote by γ(x, y) the subcurve of γ connecting y to x. We write l(γ) for the euclidean length of the curve γ.

Let f: ]0, 1[→ R. We write ∫_{0} f(t) dt = ∞, if ∫_{0}^{r} f(t) dt = ∞ for arbitrarily small r > 0.

We denote by C( ·) various positive constants that depend only on the quantities written in paranthesis.

2.1. Carathéodory’s construction. Let h be a function defined for all t ≥ 0, monotonic increasing for t ≥ 0, positive for t > 0 and continuous from the right for all t ≥ 0. Define h(G) for an open nonempty set G ∈ R^{n} by h(G) = h(d(G)), where d(G) is the diameter of G in the euclidean metric, and h(∅) = 0.

Now the set function

H^{h}(E) = lim sup_{δ→0} H_{δ}^{h}(E),

where

H_{δ}^{h}(E) = inf{∑_{i=1}^{∞} h(B_{i}) : E ⊂ ⋃_{i=1}^{∞} B_{i}, d(B_{i}) ≤ δ},

is a measure on R^{n}. It is called the Haudorff measure corresponding to the premeasure h, or simply h-measure. The construction of H^{h} is called Carathéodory’s construction in [F].

Following C. A. Rogers [R, p. 78], we write

g ≺ h,

and say that g corresponds to a smaller generalized dimension than h, if

h(t)/g(t) → 0 as t → 0.

Note also the following result (see [R, p. 79]). Let f, g, h be premeasures such that f ≺ g ≺ h. If 0 < H^{g}(E) < ∞, then H^{h}(E) = 0 and H^{f}(E) = ∞.

3. Generalized mean porosity

We define the generalized mean porosity as follows.

Definition 3.1. Let E ⊂ R^{n} be a compact set. Let α: ]0, 1[⟶]0, 1[ be a continuous function such that

(2)  \frac{α(t)}{t} is an increasing function

and let λ: Z^{+} ⟶ Z^{+} be a function. Let 𝒬 be a collection of pairwise disjoint cubes Q_{i} ⊂ R^{n} \ E. We define for each such a collection 𝒬 and for every k ∈ Z^{+} a function

χ_{k}^{𝒬} (x) = \{ \begin{matrix}1, if one can find cubes Q_{i}^{k}(x) ∈ 𝒬, i = 1, …, λ(k), \\ such that Q_{i}^{k}(x) ⊂ A_{k}(x) and l(Q_{i}^{k}(x)) ≥ α(2^{−k}) for all i; \\ 0, otherwise. \end{matrix}

Let

S_{j}^{𝒬} (x) = ∑_{k=1}^{j} χ_{k}^{𝒬} (x).

We say that a set E is weakly mean porous with parameters (α, λ), if there exists a collection 𝒬 as above and an integer j_{0} ∈ Z^{+} such that

(3)  \frac{S_{j}^{𝒬} (x)}{j} > \frac{1}{2}

for all x ∈ E and for all j ≥ j_{0} .

There are two principal differences between generalized mean porosity and other porosities. Firstly, it is not a pointwise property, since the collection 𝒬 does not depend on the point x. Secondly, it does not permit scaling, since the parameters α and λ depend on the scale. In particular, α(t)/t is not necessarily a constant (as it is in the usual mean porosity). For example, let us consider a set E ⊂ R^{2} which is weakly mean porous with parameters α(t) = ct/ log(1/t) for small t and λ(k) = k. Fix a point x ∈ E. Roughly speaking, half of the annuli A_{k}(x), k = 1, 2, …, now contain k disjoint “holes” Q_{i}^{k}(x) ∈ 𝒬 with side lengths at least c2^{−k}/k. In this section we will prove that this implies m_{2}(E) = 0 and even H^{h}(E) < ∞ for the gauge function h(t) = t^{2}(log(1/t))^{C} with some C > 0.

In Definition 3.1 the property (2) can be described as follows. We require, that as one reduces the scale, the size of the “holes” does not increase in proportion to the scale. Note that when α(t)/t is a constant and λ(k) ≡ 1 our definition is equivalent with the definition of mean porosity in [KR]. Indeed, for a mean porous set E, we can take 𝒬 to be the collection of the Whitney decompositions of all cubes in the Whitney decomposition of R^{n} \ E. The fact that this collection satisfies condition (3) is shown in the proof of [KR, Theorem 2.1].

The parameter λ(k) controls the number of “holes” in each annulus. It is important for our applications that we use a general λ that allows us to use the contribution from several “small” holes in a single set A_{k}(x) \ E.

We could also define the porosity condition of Definition 3.1 in a pointwise way (i.e. allow the collection 𝒬 to depend on the point x), as porosity conditions are defined in general. Then, however, we could prove a dimension estimate for the set E only in the case that λ(k) is bounded from above. We do not know whether it is possible to prove a sharp dimension bound for sets satisfying such a pointwise porosity condition with an unbounded parameter λ. However, in our applications we will find the collection 𝒬 independently of x, and thus Definition 3.1 works well for us.

The constant \frac{1}{2} in condition (3) plays a technical role only and could be replaced with any positive constant without essential effect on the dimension estimates. In fact, if we replace it with a constant κ > 0, then the constant C(n) in Corollary 3.5 is replaced with κC(n). Note also that our porosity condition is uniform in the sense that j_{0} is independent of x.

In order to prove a dimension estimate for weakly mean porous sets we need the following well-known consequence of the Hardy–Littlewood maximal theorem, see [Bo]. Here χ_{E} denotes the characteristic function of the set E.

Lemma 3.2. Let ℬ be a collection of balls B ⊂ R^{n} and let p ≥ 1. Then

∫_{R^{n}}( ∑_{B∈ℬ} χ_{pB}(x))^{k} dx ≤ (C_{1}kp^{n})^{k} ∫_{R^{n}}( ∑_{B∈ℬ} χ_{B}(x))^{k} dx

for all k ≥ 1, where C_{1} = C_{1}(n).

Next we introduce the main result of this section. It is an estimate on the size of weakly mean porous sets.

Theorem 3.3. There exists a positive constant C(n) such that whenever E ⊂ R^{n} is a weakly mean porous set with parameters (α, λ), then H^{h}(E) < ∞ for each premeasure h, which satisfies

(4)  h(2^{−j}) ≤ M2^{−jn} exp(C(n) inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \})

for all j > j_{0} with some positive constant M. Here the infimum is taken over all index sets I_{j} that satisfy

I_{j} = ⋃_{i=1}^{j} I_{i} with I_{i} ⊂ I_{i+1} ⊂ \{1, 2, …, i + 1\} so that \frac{♯I_{i}}{i} > \frac{1}{2} for all j_{0} ≤ i ≤ j.

Proof. Let Q_{0} = {(x_{1}, …, x_{n}) : −1 ≤ x_{j} ≤ 1}. We can assume that E is a subset of the cube Q_{0} . If this is not the case, we can subdivide E into a finite number of compact sets E_{j} so that each set fits into the cube Q_{0} . We can also assume that Q ⊂ E + 1 for all Q ∈ 𝒬.

Let j > j_{0} , and for each k ≤ j let N(k) be the smallest integer such that

N(k) ≥ \frac{2^{−j}α(2^{−k})}{2^{−k}α(2^{−j})}.

By property (2), N(k) ≥ N(k + 1). Now we define 𝒬_{j} by subdividing the cubes of the collection 𝒬 in the following way: If Q ∈ 𝒬 and there is 1 < k ≤ j such that α(2^{−k}) ≤ l(Q) < α(2^{−k+1}), then each edge of the cube Q is divided into N(k) parts. As for a cube Q with l(Q) ≥ α(2^{−1}), divide each edge into N(1) parts. Hence Q is subdivided into N(k)^{n} cubes that have edge lengths of at least 2^{−k}α(2^{−j})/2·2^{−j}. Let 𝒬_{j} be the collection of cubes acquired in this manner from the cubes Q ⊂ 𝒬 with l(Q) ≥ α(2^{−j}).

Denote the largest ball B ⊂ Q by B(Q). Let

ℬ_{j} = {B(Q) : Q ∈ 𝒬_{j}}.

Let x ∈ E + 2^{−j}. We choose x′ ∈ E such that d(x, x′) < 2^{−j}. Let k < j satisfy χ_{k}(x′) = 1. By Definition 3.1 there are cubes Q_{i} ∈ 𝒬, i = 1, …, λ(k), in the annulus A_{k}(x′) such that α(2^{−k}) ≤ l(Q_{i}). Hence from the annulus A_{k}(x′) we find disjoint balls B_{i} ∈ ℬ_{j}, i = 1, …, λ(k)N(k)^{n}, such that

r(B_{i}) ≥ \frac{1}{4} \frac{2^{−k}α(2^{−j})}{2^{−j}}.

Let I_{j} consist of all the indices k ≤ j for which χ_{k}(x′) = 1. Then, by Definition 3.1, the index set I_{j} satisfies

I_{j} = ⋃_{i=1}^{j} I_{i} with I_{i} ⊂ I_{i+1} ⊂ \{1, 2, …, i + 1\} so that \frac{♯I_{i}}{i} > \frac{1}{2} for all j_{0} ≤ i ≤ j,

where the number of indices in the set I_{i} is denoted by ♯I_{i}.

By enlarging the balls B ∈ ℬ_{j} we have that

\begin{aligned}∑_{B∈ℬ_{j}} χ_{C_{1}(n) \frac{2^{−j}}{α(2^{−j})} B}(x) & ≥ inf_{I_{j}}\{ ∑_{k∈I_{j}} λ(k)N(k)^{n}\} \\ & ≥ (\frac{2^{−j}}{α(2^{−j})})^{n} inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \}, \end{aligned}

when the constant C_{1}(n) is large enough. Hence we have the estimate

(5)  \frac{1}{G_{j}} (\frac{α(2^{−j})}{2^{−j}})^{n} ∑_{B∈ℬ_{j}} χ_{C_{1}(n) \frac{2^{−j}}{α(2^{−j})} B}(x) ≥ 1,

where

G_{j} = inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \}.

Next we use inequality (5) to estimate the Lebesgue measure of a neighborhood of E. For all 0 < t < 1 and Q > 0 we have that

|E + 2^{−j}| exp(\frac{G_{j}}{Q} ) ≤ ∫_{E+2^{−j}} ∑_{i≥0} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} dx

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} )

+ ∑_{i≥1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} ∫_{R^{n}}(\frac{1}{G_{j}} (\frac{α(2^{−j})}{2^{−j}})^{n}

× ∑_{B∈ℬ_{j}} χ_{C_{1}(n) \frac{2^{−j}}{α(2^{−j})} B}(x))^{ti} dx.

By combining Lemma 3.2, inequality i^{i} ≤ e^{i}i! and Hölder’s inequality we thus deduce that

|E + 2^{−j}| exp(\frac{G_{j}}{Q} ) ≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} )

+ ∑_{1/t≤i} \frac{1}{i!} \frac{G_{j}^{(1−t)i}}{Q^{i}} (C_{2}(n)tiC_{1}(n)^{n}(\frac{α(2^{−j})}{2^{−j}})^{n}

× (\frac{2^{−j}}{α(2^{−j})})^{n})^{ti} ∫_{R^{n}}( ∑_{B∈ℬ_{j}} χ_{B}(x))^{ti}

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} + ∑_{1/t≤i} \frac{G_{j}^{(1−t)i}(C_{3}(n)ti)^{ti}}{Q^{i}i!} )

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} + ∑_{1/t≤i} \frac{G_{j}^{(1−t)i}(i!e^{i})^{t}(C_{3}(n)t)^{ti}}{Q^{i}i!} )

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} + ∑_{1/t≤i} \frac{t^{ti}G_{j}^{(1−t)i}(C_{3}(n)e)^{ti}}{Q^{i}(i!)^{1−t}} )

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} + ( ∑_{1/t≤i} t^{ti/t})^{t}

× ( ∑_{1/t≤i} \frac{G_{j}^{i}(C_{3}(n)e)^{ti/(1−t)}}{Q^{i/(1−t)}i!} )^{1−t})

≤ |E + 1|( ∑_{0≤i<1/t} \frac{1}{i!} \frac{G_{j}^{i}}{Q^{i}} + (\frac{1}{1 − t})^{t}

× exp(G_{j}(C_{3}(n)e)^{t/(1−t)}(\frac{1 − t}{Q^{1/(1−t)}} )))

≤ M_{0}(n) exp(\frac{G_{j}}{2Q}),

when we choose t = \frac{1}{2} , a constant M_{0}(n) big enough and a constant Q = C_{3}(n)e. By the previous calculations we arrive at

|E + 2^{−j}| exp(\frac{G_{j}}{2Q}) ≤ M_{0}(n),

and hence

(6)  |E + 2^{−j}| exp(C(n)G_{j}) ≤ M_{0}(n),

where C(n) = 1/2C_{3}(n)e.

The desired dimension estimate follows from inequality (6) by a standard calculation using the Besicovich covering theorem. We show this in the following.

Let 𝒜 be the collection of all the balls of radii 2^{−j} with centers in the set E. By the Besicovich covering theorem we can choose balls B_{i} ∈ 𝒜 , i = 1, …, m_{j}, such that E ⊂ ⋃_{i=1}^{m_{j}} B_{i} and

(7)  ∑_{i=1}^{m_{j}} χ_{B_{i}}(x) < P (n)

for all x ∈ R^{n}. By (6) and (7) we have that

\frac{M_{0}(n)}{exp(C(n)G_{j})} ≥ |E + 2^{−j}| ≥ m_{j}Ω_{n}(2^{−j})^{n} \frac{1}{P (n)},

and hence

m_{j} ≤ \frac{M_{0}(n)P (n)2^{jn}}{Ω_{n} exp(C(n)G_{j})}.

Let h be a premeasure satisfying (4). Then we obtain the estimate

H^{h}(E) ≤ lim sup_{j→∞}{m_{j}h(2^{−j})}

≤ lim sup_{j→∞}{m_{j}M2^{−jn} exp(C(n)G_{j})}

≤ lim sup_{j→∞}{R(n)M2^{jn} exp(−C(n)G_{j})2^{−jn} exp(C(n)G_{j})}

≤ R(n)M < ∞.

Let us make some remarks on Theorem 3.3. Notice that λ(k)α(2^{−k})^{n} estimates the total volume of the cubes or “holes” Q_{i}^{k}(x) ∈ 𝒬, i = 1, …, λ(k), in the annulus A_{k}(x), where k satisfies χ_{k}^{𝒬} (x) = 1. The index set I_{j} consists of all indices k ≤ j for which χ_{k}^{𝒬} (x) = 1. Hence I_{j} depends on the point x. That is why the infimum is taken over all possible index sets. The condition given for I_{j} is implied directly by Definition 3.1.

Note the following special cases of Theorem 3.3. If we have for arbitrarily large j that

G_{j} = inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \} ≥ Cj

with some constant C > 0, then it follows from Theorem 3.3 that dim_{H}(E) < n. Note that this happens, for example, if we have the parameters α(t) = ct and λ(k) ≡ 1, in other words, if the set E is mean porous.

If G_{j} → ∞ as j → ∞, then m_{n}(E) = 0, and Theorem 3.3 will also give us a dimension estimate with the gauge function h. However, if G_{j} is bounded, i.e. there is M ∈ R such that G_{j} < M for all j, then Theorem 3.3 does not give us a dimension estimate. Indeed, in this case the set E can have positive Lebesgue measure, see Section 7.1.

Let us also point out that, in fact, we proved more than what we claim in Theorem 3.3. Indeed, we proved inequality (6), which is a stronger condition for the set E than the claimed dimension estimate.

In the next remark we show that in certain cases the index set I_{j} of Theorem 3.3 can be given explicitly.

Remark 3.4. If it holds for the parameters in Theorem 3.3 that

(8)  p(k) := \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}}

is increasing as a function of k, then (for even j)

inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \} = ∑_{k=1}^{j/2} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} ≥ Cj

with some positive constant C.

If however p(k) is decreasing as a function of k, then (for even j_{0} )

inf_{I_{j}}\{ ∑_{k∈I_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} \} = ∑_{k∈J_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} ,

where

J_{j} = \{\frac{j_{0}}{2} + 1, \frac{j_{0}}{2} + 2, …, j_{0}\} ∪ \{i ∈ \{j_{0} + 1, …, j\} such that i is odd\}.

Moreover, for all j > j_{0} we have that

∑_{k∈J_{j}} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} ≥ \frac{1}{2} ∑_{k=j_{0}}^{j} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} .

Corollary 3.5. Let E ⊂ R^{n} be a weakly mean porous set with parameters (α, λ) such that p(k) (defined by (8)) is a decreasing function of k and

∑_{k=j_{0}}^{∞} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} = ∞.

Then m_{n}(E) = 0 and there exists a positive constant C(n) such that H^{h}(E) < ∞ for each premeasure h, which satisfies

h(2^{−j}) ≤ M2^{−jn} exp(C(n) ∑_{k=j_{0}}^{j} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} )

for all j > j_{0} with some positive constant M.

Proof. The claim follows by combining Theorem 3.3 and Remark 3.4.

Note that this corollary is essentially sharp by an example in Section 7.1.

4. A quasihyperbolic growth condition

Let Ω ⊂ R^{n} be a domain. We recall that the quasihyperbolic distance between two points x_{1}, x_{2} ∈ Ω is defined as

k_{Ω}(x_{1}, x_{2}) = inf_{γ} ∫_{γ} \frac{ds}{d(x, ∂Ω)}

where the infimum is taken over all rectifiable arcs joining x_{1} to x_{2} in Ω.

Definition 4.1. Let ϕ: ]0, 1] ⟶]0, ∞[ be a continuous and decreasing function. We say that a bounded domain Ω ⊂ R^{n} satisfies a quasihyperbolic growth condition with a function ϕ, if there is a point x_{0} ∈ Ω and a constant C_{0} such that

(9)  k_{Ω}(x_{0}, x) ≤ ϕ(\frac{d(x, ∂Ω)}{d(x_{0}, ∂Ω)}) + C_{0}

for all x ∈ Ω.

Note that for a bounded domain we can always choose the point x_{0} so that d(x, ∂Ω)/d(x_{0}, ∂Ω) ≤ 1 for all x ∈ Ω, and hence the domain of ϕ can be assumed to be ]0, 1]. Recall that if a domain Ω satisfies condition (9) with a function ϕ(t) = C log(1/t), then Ω is called a Hölder domain (see [SS1]). Thus we can say that domains defined in Definition 4.1 are generalized Hölder domains. It is well known that the Hausdorff dimension of the boundary of a Hölder domain is strictly smaller than n. This is shown in [JM], [SS2] and [KR]. In this section we prove a corresponding dimension estimate for domains satisfying (9) with a function ϕ that satisfies certain conditions formulated below. To indicate how fast decreasing functions ϕ allow for a generalized dimension estimate, let us already point out that, for ϕ(t) = (log(1/t))^{s} we will obtain a dimension bound when s ≤ n/(n − 1), whereas the boundary can have positive volume when s > n/(n − 1).

Definition 4.2. We say that a decreasing function ϕ: ]0, 1] →]0, ∞[ is of logarithmic type, if there exist positive constants t_{0} < 1 and β such that ϕ satisfies the following conditions for all t ≤ t_{0} :

(10)  ϕ(t) is differentiable and − ϕ′(t)t is a decreasing function;

(11)  ϕ(t) ≤ βϕ(\sqrt{t} ).

Note that, for example, a function of the form

ϕ(t) = \{ \begin{matrix}C(log \frac{1}{t})^{s_{1}}(log log \frac{1}{t})^{s_{2}} ⋯ (log^{(m)} \frac{1}{t})^{s_{m}} + C, & t < a_{m}; \\ C, & t ≥ a_{m}, \end{matrix}

where C > 0, m ∈ Z^{+} , a_{m} = 1/ exp^{(m−1)}(e), s_{1} ≥ 1, s_{2}, …, s_{m} ≥ 0, is of logarithmic type.

Lemma 4.3. Let ϕ be a function of logarithmic type. Then

ϕ(ab) ≤ β(ϕ(a) + ϕ(b))

for all a, b ∈]0, 1[ for which ab ≤ t_{0} .

Proof. Either a ≤ \sqrt{ab} or b ≤ \sqrt{ab} , and hence we obtain β(ϕ(a) + ϕ(b)) ≥ β max\{ϕ(a), ϕ(b)\} ≥ βϕ(\sqrt{ab} ) ≥ ϕ(ab).

Lemma 4.4. Let ϕ be a function of logarithmic type. Then there is t_{1} ∈]0, 1[ such that the inequality

βϕ(t^{k+1}) ≤ 2^{−k}(\frac{1}{t})^{k}

holds for all t < t_{1} and every k ∈ Z^{+} .

Proof. Let t_{0} < 1 be as in Definition 4.2. We show first that there is t̃_{1} such that

(12)  ϕ(t) ≤ \frac{1}{t}

for all t < t̃_{1} . Suppose that (12) is false. Then for arbitrarily large j ∈ Z^{+} there is t_{0}^{2} ≤ r_{j} ≤ t_{0} such that ϕ(r_{j}^{2^{j}}) > (1/r_{j})^{2^{j}} ≥ (1/t_{0})^{2^{j}}. By iterating condition (11), we obtain ϕ(r_{j}^{2^{j}}) ≤ β^{j}ϕ(r_{j}) ≤ β^{j+1}ϕ(t_{0}), and hence (1/t_{0})^{2^{j}} < β^{j+1}ϕ(t_{0}). This is a contradiction with large j, and thus property (12) is proved.

Let k ∈ Z^{+} . Applying property (11) twice, we have that

βϕ(t^{k+1}) ≤ β^{3}ϕ(t^{(k+1)/4})

for all t < t_{0}^{2} . Then, by property (12), we obtain

β^{3}ϕ(t^{(k+1)/4}) ≤ β^{3}(\frac{1}{t})^{(k+1)/4}

for all t < t̃_{1}^{2} . A simple calculation yields

β^{3}(\frac{1}{t})^{(k+1)/4} ≤ 2^{−k}(\frac{1}{t})^{k} for all t < \frac{1}{4β^{6}} .

This proves the claimed inequality for all t < t_{1} = min{t_{0}^{2}, t̃_{1}^{2}, 1/4β^{6}}.

The next theorem extends a result by Smith and Stegenga in [SS1, Theorem 3] given for Hölder domains. For an intermediate result see [KOT, Lemma 4.6].

Theorem 4.5. Let Ω ⊂ R^{n} be a bounded domain that satisfies the quasihyperbolic growth condition with the function ϕ of logarithmic type. Then there is a constant C_{Ω} < ∞ such that

(13)  k_{Ω}(x, x_{0}) ≤ βϕ(\frac{l(γ(x, x_{1}))}{d(x_{0}, ∂Ω)}) + C_{Ω}

for all x_{1} ∈ Ω, where γ is a quasihyperbolic geodesic connecting x_{0} to x_{1} , and x ∈ γ.

Proof. Assume that (13) is false. Then for each constant C_{Ω} there is a point x_{1} , a geodesic γ connecting x_{0} to x_{1} , and a point y_{0} ∈ γ for which

(14)  βϕ(\frac{l(γ(y_{0}, x_{1}))}{d(x_{0}, ∂Ω)}) + C_{Ω} < k_{Ω}(x_{0}, y_{0}).

Let L = l(γ(y_{0}, x_{1})). Define points y_{k} ∈ γ(y_{k−1}, x_{1}) recursively so that l(γ(y_{k−1}, y_{k})) = 2^{−k}L for all k ∈ Z^{+} . Let

δ_{k} = sup{d(x, ∂Ω) : x ∈ γ(y_{k}, x_{1})}.

We can choose the constant C_{Ω} so large that δ_{0}/d(x_{0}, ∂Ω) ≤ t_{0} . Then, by combining (9), (14) and Lemma 4.3, we obtain the following chain of inequalities for all x ∈ γ(y_{0}, x_{1}):

βϕ(\frac{L}{d(x_{0}, ∂Ω)}) + C_{Ω} < k_{Ω}(x_{0}, y_{0})

≤ k_{Ω}(x_{0}, x) ≤ ϕ(\frac{d(x, ∂Ω)}{d(x_{0}, ∂Ω)}) + C_{0}

≤ βϕ(\frac{d(x, ∂Ω)}{L}) + βϕ(\frac{L}{d(x_{0}, ∂Ω)}) + C_{0}.

Hence

C_{Ω} − C_{0} ≤ βϕ(\frac{δ_{0}}{L} ).

Now we can choose the constant C_{Ω} so large that C_{Ω} ≥ C_{0} and the ratio δ_{0}/L is so small that, by Lemma 4.4,

(15)  βϕ((\frac{δ_{0}}{L} )^{k+1}) ≤ 2^{−k}(\frac{L}{δ_{0}} )^{k}

for all k ∈ Z^{+} .

We show by induction that δ_{k−1}/L ≤ (δ_{0}/L)^{k} for all k ∈ Z^{+} . This is trivially true if k = 1, so assume that it is true for some k ≥ 1. By combining the induction assumption, Lemma 4.3 and the inequalities (14) and (15), we obtain for all x ∈ γ(y_{k}, x_{1}) that

βϕ(\frac{L}{d(x_{0}, ∂Ω)}) + C_{Ω} + βϕ((\frac{δ_{0}}{L} )^{k+1}) ≤ k_{Ω}(x_{0}, y_{0}) + 2^{−k}(\frac{L}{δ_{0}} )^{k}

≤ k_{Ω}(x_{0}, y_{0}) + 2^{−k} \frac{L}{δ_{k−1}}

≤ k_{Ω}(x_{0}, y_{0}) + k_{Ω}(y_{k−1}, y_{k}) ≤ k_{Ω}(x_{0}, x)

≤ ϕ(\frac{d(x, ∂Ω)}{d(x_{0}, ∂Ω)}) + C_{0}

≤ βϕ(\frac{d(x, ∂Ω)}{L})+ βϕ(\frac{L}{d(x_{0}, ∂Ω)})+ C_{0}.

Now we have that

βϕ((δ_{0}/L)^{k+1}) + C_{Ω} − C_{0} ≤ βϕ(\frac{δ_{k}}{L} )

which proves the induction hypothesis.

Since δ_{0}/L < 1 and the inequality

0 < d(x_{1}, ∂Ω) ≤ δ_{k} ≤ L(\frac{δ_{0}}{L} )^{k+1}

holds for all k ∈ Z^{+} , we have a contradiction which proves the theorem.

For the proof of the main theorem of this section we need one more lemma concerning the geometric properties of the Whitney decomposition. For the exact construction of this decomposition we refer the reader to [S].

Lemma 4.6. Let Q_{0} ⊂ R^{n} be a cube that has sides parallel to the coordinate planes, and let the edge length of Q_{0} be 2^{−m}. Let Q̃ ⊂ Q_{0} be a cube sharing a part of a face with Q_{0} . Let l(Q̃) = c2^{−m} with c < 1. Let 𝒲 be a Whitney decomposition of Q_{0} . Then, there is a cube Q ∈ 𝒲 for which Q ⊂ Q̃ and l(Q) ≥ c2^{−m}/D(n). Moreover, there are at least 2^{i(n−1)} cubes Q_{j} ∈ 𝒲 for which Q_{j} ⊂ Q̃ and l(Q_{j}) ≥ c2^{−m−i}/D(n). Here D(n) = 2 + 6\sqrt{n} .

Proof. Recall that each Whitney cube Q_{j}^{k} ∈ 𝒲 has sides parallel to the coordinate planes and the edge length of Q_{j}^{k} is 2^{−k}. The collection {Q_{j}^{k} : j = 1, …, N_{k}} is called the k^{th} generation of the cubes. It follows from the construction of the Whitney decomposition that the inequality

(16)  2^{−k}\sqrt{n} ≤ d(Q_{j}^{k}, ∂Q_{0}) < 3 · 2^{−k}\sqrt{n}

holds for each cube in the k^{th} generation, see [G]. Thus we see that there must be a cube Q ∈ 𝒲 such that Q ⊂ Q̃ and

l(Q) ≥ \frac{c2^{−m}}{2 + 6\sqrt{n}}

as otherwise the inequality (16) would fail for some cube near the center of Q̃ .

To prove the second part of the lemma let i ∈ Z^{+} and subdivide the cube Q̃ into 2^{in} cubes with equal side lengths of at least c2^{−m−i}. Of these subcubes at least 2^{i(n−1)} cubes share a face with the cube Q_{0} and, by inequality (16), from each subcube we find a cube Q_{j} ∈ 𝒲 such that l(Q_{j}) ≥ c2^{−m−i}/D(n).

We recall also the following property of the Whitney decomposition. Let 𝒲 be the Whitney decomposition of a domain Ω ⊂ R^{n}. Pick a cube Q_{0} ∈ 𝒲 , and set q(Q_{0}) = 0. For any two adjacent (i.e. sharing at least a part of a face) Whitney cubes, join their centers by an interval, and let q(Q) be the number of intervals in the shortest chain joining the centers of Q_{0} and Q. We can remove the redundant intervals so that the resulting collection of intervals is a tree. We denote the set of cubes connecting Q to Q_{0} by chain(Q_{0}, Q), and the number of cubes in chain(Q_{0}, Q) by ♯chain(Q_{0}, Q). Note that now q(Q) + 1 = ♯chain(Q_{0}, Q) ≤ Ck_{Ω}(z_{0}, z) for any z_{0} ∈ Q_{0} and z ∈ Q for which k_{Ω}(z_{0}, z) > constant.

The next theorem extends the result given for Hölder domains in [KR, Theorem 5.1]. We show that the boundary of a generalized Hölder domain is weakly mean porous with appropriate parameters.

Theorem 4.7. Let Ω ⊂ R^{n} be a bounded domain that satisfies the quasihyperbolic growth condition with the function ϕ of logarithmic type. Then there is a constant c > 0 such that the boundary of the domain Ω is weakly mean porous with parameters C_{1}(n)α(t) and C_{2}(n)λ(k), where (for small t) α(t) = c/−ϕ′(t) and λ(k) is the smallest integer such that λ(k) ≥ 2^{−k}/α(2^{−k}).

Proof. Let j_{0} ∈ Z^{+} such that 2^{−j_{0}+1} ≤ d(x_{0}, ∂Ω). Let j > 2j_{0} and let x ∈ ∂Ω. Choose a point

y ∈ B(x, 2^{−j−1}) ∩ Ω,

and let γ be a quasihyperbolic geodesic connecting y to x_{0} . Choose w ∈ γ such that

l(γ(w, y)) = 2^{−j−1}.

Then w ∈ B(x, 2^{−j}). Moreover, Lemma 4.5 implies the estimate

(17)  k_{Ω}(w, x_{0}) ≤ βϕ(\frac{2^{−j−1}}{d(x_{0}, ∂Ω)}) + C_{Ω}.

Define for each k ≥ j_{0} a function

χ_{k}(x) = \{ \begin{matrix}1, & if ∫_{A_{k}(x)∩γ} \frac{dt}{d(t, ∂Ω)} ≤ \frac{2^{−k}}{α(2^{−k})}; \\ 0, & otherwise. \end{matrix}

Let

S_{j}(x) = ∑_{k=j_{0}}^{j} χ_{k}(x).

We prove first that

(18)  \frac{S_{j}(x)}{j} > \frac{1}{2}

for all sufficiently large j.

For those A_{k}(x), j_{0} ≤ k ≤ j, for which χ_{k}(x) = 0, we have that

(19)  ∫_{A_{k}(x)∩γ} \frac{dt}{d(t, ∂Ω)} > \frac{2^{−k}}{α(2^{−k})}.

Suppose that the assertion S_{j}(x)/j > \frac{1}{2} fails for some large j. Then by (19) we have that

k_{Ω}(w, x_{0}) > ∑_{k=j_{0}}^{j} |χ_{k}(x) − 1|\frac{2^{−k}}{α(2^{−k})},

which is by property (10) at least

∑_{k=j_{0}}^{j/2} \frac{−ϕ′(2^{−k})2^{−k}}{c} ≥ \frac{1}{c}(ϕ(2^{−j/2}) − ϕ(2^{−j_{0}})).

This number is greater than βϕ(2^{−j−1}/d(x_{0}, ∂Ω)) + C_{Ω} , when we choose j big enough and c < 1/β^{4} . Hence we have a contradiction with inequality (17), which proves (18).

Next we define a collection 𝒬 of disjoint cubes in Ω in the following way. Let 𝒲 be a Whitney decomposition of the domain Ω. Then let 𝒬 consist of all the cubes in the Whitney decompositions of the cubes Q ∈ 𝒲 . We show that

(20)  χ_{k}^{𝒬} (x) ≥ χ_{k}(x)

for all k ≥ j_{0} with parameters C_{1}(n)α(2^{−k}) and C_{2}(n)λ(k).

Consider k ∈ Z^{+} such that χ_{k}(x) = 1. Then

∫_{A_{k}(x)∩γ} \frac{dt}{d(t, ∂Ω)} ≤ \frac{2^{−k}}{α(2^{−k})}.

Choose y ∈ γ ∩ S^{n−1}(x, 2^{−k+1}) and z ∈ γ ∩ S^{n−1}(x, 2^{−k}) such that γ(y, z) ⊂ A_{k}(x). Let Q_{y}, Q_{z} ∈ 𝒲 such that y ∈ Q_{y} and z ∈ Q_{z}. Now

(21)  ♯chain(Q_{y}, Q_{z}) ≤ \frac{c_{0}2^{−k}}{α(2^{−k})}

with some constant c_{0} depending only on n.

For each Q_{i} ∈ chain(Q_{y}, Q_{z}), let Q̃_{i} ⊂ R^{n} be the largest cube such that it has sides parallel to the coordinate planes and Q̃_{i} ⊂ Q_{i} ∩ \overline{A_{k}(x)}. Now Q̃_{i} shares at least one part of a face with Q_{i}. Moreover

(22)  ∑_{i=1}^{♯chain(Q_{y},Q_{z})} d(Q̃_{i}) ≥ 2^{−k}.

Combining (21) and (22) we have that

(23)  ∑_{i:d(_{Q}_{i})≥α(2^{−k})/2c_{0}} d(Q̃_{i}) ≥ \frac{2^{−k}}{2} .

Applying Lemma 4.6 and inequality (23) we see that from these cubes Q̃_{i}, for which d(Q̃_{i}) ≥ α(2^{−k})/2c_{0} , we find at least c_{0}2^{−k}/α(2^{−k}) cubes Q ∈ 𝒬 such that l(Q) ≥ α(2^{−k})/2c_{0}D(n)\sqrt{n} . Thus we have proven (20) with constants C_{1}(n) = 1/2c_{0}D(n)\sqrt{n} and C_{2}(n) = c_{0}/2. The claim follows immediately from (18) and (20).

Note that property (10) for the function ϕ implies property (2) in Definition 3.1 for the function α(t) = c/−ϕ′(t). Also note that, for a Hölder domain, −ϕ′(t)t is a constant, and by Theorem 4.7 the boundary of such a domain is mean porous (this result is equivalent to [KR, Theorem 5.1]).

Corollary 4.8. Let Ω ⊂ R^{n} be a bounded domain that satisfies the quasihyperbolic growth condition with the function ϕ of logarithmic type. Then there are positive constants M, C(β, n) and an integer j_{0} such that

|∂Ω + 2^{−j}| ≤ M exp(−C(β, n) ∫_{[2^{−j},2^{−j_{0}} ]} \frac{dt}{(−ϕ′(t)t)^{n−1}t})

for all j > j_{0} .

Proof. By combining Theorem 4.7 and the proof of Theorem 3.3 we deduce by (6) that there are j_{0} ∈ Z^{+} and c = c(β) > 0 such that (the summation indices follow from Remark 3.4)

\begin{aligned}|∂Ω + 2^{−j}| & ≤ M exp(−C(n) ∑_{k=j_{0}}^{j}(\frac{c}{−ϕ′(2^{−k})2^{−k}} )^{n−1}) \\ & ≤ M exp(−\frac{1}{2}C(n)c^{n−1} ∫_{[2^{−j},2^{−j_{0}} ]} \frac{dt}{(−ϕ′(t)t)^{n−1}t}) \end{aligned}

for all j > j_{0} with a constant M depending on Ω.

Corollary 4.9. Let Ω ⊂ R^{n} be a bounded domain that satisfies the quasihyperbolic growth condition with the function ϕ of logarithmic type satisfying

∫_{0} \frac{dt}{(−ϕ′(t)t)^{n−1}t} = ∞.

Then m_{n}(∂Ω) = 0 and there is a positive constant C(β, n) and an integer j_{0} such that H^{h}(∂Ω) < ∞ for each premeasure h, which satisfies

h(2^{−j}) ≤ M2^{−jn} exp(C(β, n) ∫_{[2^{−j},2^{−j_{0}} ]} \frac{dt}{(−ϕ′(t)t)^{n−1}t})

for all j > j_{0} with some positive constant M.

Proof. The claim follows by Theorem 4.7, Corollary 3.5 and a similar argument as in the proof of Corollary 4.8.

Note that Corollary 4.9 is essentially sharp by an example in Section 7.2.

Remark 4.10. Let Ω ⊂ R^{n} be a bounded domain that satisfies the quasihyperbolic growth condition with the function

ϕ(t) = \frac{1}{ε}(log \frac{1}{t})^{s} with 1 ≤ s ≤ \frac{n}{n − 1}.

Then, by Corollary 4.9, m_{n}(∂Ω) = 0 and H^{h}(∂Ω) < ∞ for the gauge function

h(t) = t^{n} exp(\frac{C}{(n − (n − 1)s)}(log \frac{1}{t})^{n−(n−1)s})

when s < n/(n − 1), and for the gauge function

h(t) = t^{n}(log \frac{1}{t})^{C}

when s = n/(n − 1). Here the constant C depends on ε, n and s.

If n/(n − 1) < s, then the boundary of the domain Ω can have positive Lebesgue measure, see Section 7.2.

5. Uniform continuity of quasiconformal mappings

The connection between uniform continuity of quasiconformal mappings and the concept of generalized mean porosity comes from the following observation.

Theorem 5.1. Let ψ: ]0, 1[→]0, 1[ be an increasing bijection, and let u(t) := ψ^{−1}(t). Assume that log(1/u(t)) is of logarithmic type. Let f: B^{n} → Ω ⊂ R^{n} be a K -quasiconformal map such that the inequality

(24)  |f(tω) − f(ω)| ≤ ψ(1 − t)

holds for all ω ∈ S^{n−1} and t_{0} < t < 1. Then there is a constant c > 0 such that ∂f(B^{n}) is weakly mean porous with parameters C_{1}(n)α(t) and C_{2}(n)λ(k), where (for small t) α(t) = cu(t)/u′(t) and λ(k) is the smallest integer such that λ(k) ≥ 2^{−k}/α(2^{−k}).

Let us already remark that this theorem could be considered as a special case of Theorem 4.7. At the end of this section we discuss the connection between the uniform continuity of a quasiconformal mapping f and the quasihyperbolic growth condition in the image domain f(B^{n}).

Proof of Theorem 5.1. Let ω ∈ ∂B^{n}. Define functions χ_{k} and S_{j} as in the proof of Theorem 4.7. We prove that

(25)  \frac{S_{j}(f(ω))}{j} > \frac{1}{2}

for all sufficiently large j ∈ Z^{+} .

Let j_{0} ∈ Z^{+} such that 2^{−j_{0}+1} ≤ d(f(0), ∂f(B^{n})) and 2^{−j_{0}+1} ≤ 1 − t_{0} . Let j > 2j_{0} and let j_{0} ≤ k ≤ j such that χ_{k}(f(ω)) = 0. The curve γ = f([0, ω]) intersects the two boundary components of A_{k}(f(ω)) in two points a = f(t_{a}ω) and b = f(t_{b}ω), say. The quasihyperbolic distance k_{Ω}(a, b) of a and b is at least 2^{−k}/α(2^{−k}). As quasiconformal maps are quasi-isometries for large distances in the quasihyperbolic metrics (see [GO, p. 62]), the quasihyperbolic distance k_{B^{n}}(t_{a}ω, t_{b}ω) is at least C2^{−k}/α(2^{−k}), provided c is small enough. Here C depends on K and n.

Consider the largest t < 1 with

|f(tω) − f(ω)| = 2^{−j}.

It follows from (24) that 2^{−j} ≤ ψ(1 − t), and equivalently

(26)  log \frac{1}{1 − t} ≤ log(\frac{1}{u(2^{−j})}).

On the other hand

(27)  log \frac{1}{1 − t} = k_{B^{n}}(0, tω) ≥ ∑ k_{B^{n}}(t_{a}ω, t_{b}ω) ≥ ∑ C \frac{2^{−k}}{α(2^{−k})},

where the summation is over all j_{0} ≤ k ≤ j with χ_{k}(f(ω)) = 0.

Suppose that the assertion S_{j}(f(ω))/j > 1/2 fails for some large j. Then, by combining (26) and (27), we obtain that (the summation indices follow from assumption (10))

log(\frac{1}{u(2^{−j})}) ≥ C ∑_{k=j_{0}}^{j/2} \frac{2^{−k}u′(2^{−k})}{cu(2^{−k})}

≥ C \frac{1}{c}(log(\frac{1}{u(2^{−j/2})}) − log(\frac{1}{u(2^{−j_{0}})})).

This contradicts property (11) when we choose j large enough and the constant c < C/2β, and thus (25) is proved. To prove the claim we can choose the collection 𝒬 and the constants C_{1} , C_{2} similarly as in the proof of Theorem 4.7.

Corollary 5.2. Let f: B^{n} → Ω ⊂ R^{n} be a K -quasiconformal map and suppose

|f(x) − f(x′)| ≤ ψ(|x − x′|)

for all x, x′ ∈ B^{n} sufficiently close to each other. Assume that the function log(1/ψ^{−1}) is of logarithmic type and that u = ψ^{−1} satisfies

(28)  ∫_{0}(\frac{u(t)}{u′(t)})^{n−1} \frac{dt}{t^{n}} = ∞.

Then m_{n}(∂f(B^{n})) = 0 and there is a positive constant C = C(β, K, n) and an integer j_{0} such that H^{h}(∂f(B^{n})) < ∞ for each premeasure h which satisfies

h(2^{−j}) ≤ M2^{−jn} exp(C ∫_{[2^{−j},2^{−j_{0}} ]}(\frac{u(t)}{u′(t)})^{n−1} \frac{dt}{t^{n}} )

for all j > j_{0} with some positive constant M.

Proof. The claim follows by combining Theorem 5.1, Corollary 3.5, and a similar argument as in the proof of Corollary 4.8.

Remark 5.3. Consider (28) in the case n = 2. Assume that u = ψ^{−1} satisfies the conditions of Corollary 5.2 and

(29)  \frac{(log ψ(t))′}{log ψ(t)} t log t is monotone for all sufficiently small t.

Then

∫_{0}(\frac{u(t)}{u′(t)})\frac{dt}{t^{2}} = ∞ if and only if ∫_{0}(\frac{log ψ(t)}{log t})^{2} \frac{dt}{t} = ∞.

The statement above seems to be known (cf. [JM, p. 453]), but for the convenience of the reader we present a proof.

Proof of Remark 5.3. By a change of variable we have for arbitrarily small r > 0 that

∫_{0}^{r}(\frac{u(t)}{u′(t)}) \frac{dt}{t^{2}} = ∫_{0}^{ψ^{−1}(r)}(\frac{ψ′(u)}{ψ(u)})^{2}u du

= ∫_{0}^{ψ^{−1}(r)}((log ψ(u))′)^{2}u du

= ∫_{0}^{ψ^{−1}(r)}(\frac{(log ψ(u))′}{(log u)′} )^{2} \frac{du}{u}.

Let us write ψ(t) = exp(−α(t)) and assume that log(1/ψ^{−1}) is of logarithmic type (as assumed in Corollary 5.2). By (29) it suffices to show that

ε ≤ |\frac{(log ψ(t))′}{(log t)′} |/ | \frac{log ψ(t)}{log t}| = |\frac{α′(t)}{α(t)}t log t| ≤ M

for some arbitrarily small t > 0 with some positive constants ε and M. Assume towards a contradiction that

|\frac{α′(t)}{α(t)}t log t| < ε

for all small t with arbitrarily small ε > 0. By Gronwall’s lemma ([W, p. 436]) we deduce that α(t) ≤ C(log(1/t))^{ε} for small t and hence

ψ(t) ≥ exp(−C(log \frac{1}{t})^{ε}).

This implies that

ψ^{−1}(t) ≤ exp(−(\frac{1}{C} log \frac{1}{t})^{1/ε})

or equivalently

log \frac{1}{ψ^{−1}(t)} ≥ (\frac{1}{C} log \frac{1}{t})^{1/ε}

for small t. But this is a contradiction with (11) when ε is chosen small enough. Let us then assume that |α′(t)/α(t)t log t| > M for all small t with arbitrarily large M. Again by Gronwall we deduce that α(t) ≥ C(log(1/t))^{M} and hence ψ(t) ≤ exp(−C(log(1/t))^{M}). This implies that

ψ^{−1}(r) ≥ exp(−(\frac{1}{C} log \frac{1}{t})^{1/M})

or equivalently

(\frac{1}{C} log \frac{1}{t})^{1/M} ≥ log(\frac{1}{ψ^{−1}(t)}).

This, however, contradicts property (10) when M is chosen large enough.

By Remark 5.3 we see that, in the case n = 2, condition (28) is essentially equivalent with the assumption of [JM, Theorem C.1]. Jones and Makarov proved in this paper that this condition, implying m_{2}(∂f(B^{2})) = 0, is sharp. We will show in Section 7.2 that even the dimension estimate of Corollary 5.2 is essentially sharp.

Remark 5.4. Let f: B^{n} → Ω ⊂ R^{n} be a K -quasiconformal map and suppose

|f(x) − f(x′)| ≤ ψ(|x − x′|)

for all x, x′ ∈ B^{n} with the function

ψ(t) = exp(−(ε log \frac{1}{t})^{1/s})

where 1 ≤ s ≤ n/(n − 1). Then, by Corollary 5.2, m_{n}(∂f(B^{n})) = 0 and H^{h}(∂f(B^{n})) < ∞ for the gauge function

h(t) = t^{n} exp(\frac{C}{(n − (n − 1)s)}(log \frac{1}{t})^{n−(n−1)s})

when s < n/(n − 1), and for the gauge function

h(t) = t^{n}(log \frac{1}{t})^{C}

when s = n/(n − 1). Here the constant C depends on K, ε, n and s.

If n/(n − 1) < s, then the boundary of the domain f(B^{n}) can have positive Lebesgue measure, see Section 7.2.

Note that the previous example is roughly equivalent with Remark 4.10. Indeed, by using the fact that quasiconformal mappings are quasi-isometries for large distances in the quasihyperbolic metrics, we see the following connection between uniform continuity of quasiconformal mappings and the quasihyperbolic growth condition. If f: B^{n} → Ω ⊂ R^{n} is a K -quasiconformal mapping from the unit ball onto a bounded domain Ω, and

|f(x) − f(x′)| ≤ ψ(|x − x′|)

for all x, x′ ∈ B^{n} sufficiently close to each other, where ψ satisfies the conditions of Theorem 5.1, then the image domain f(B^{n}) satisfies the quasihyperbolic growth condition with the function ϕ(t) = C log(1/ψ^{−1}(t)). Moreover, the dimension estimates implied by Corollaries 4.9 and 5.2 for the boundary of f(B^{n}) are equivalent (except perhaps with different constants).

6. John domains

Definition 6.1. Let φ: [0, ∞[→ [0, ∞[ be a continuous function such that φ(t)/t is an increasing function. We say that a domain Ω is a φ-John domain, if there is a John center x_{0} ∈ Ω such that for all x ∈ Ω there is a curve γ: [0, l] → Ω, parametrized by arclength and with γ(0) = x, γ(l) = x_{0} , and d(γ(t), ∂Ω) ≥ φ(t) for all 0 < t < l.

Note that, when φ(t) = ct with some c < 1, this definition is equivalent to the definition of a usual c-John domain. The Hausdorff dimension of the boundary of a usual c-John domain is known to be strictly less than n, see e.g. [KR, p. 599]. The question arises, whether one could establish a dimension bound for the boundary of a φ-John domain Ω ⊂ R^{n} with φ(t) = ct^{s} for some s > 1. This cannot be done, however. In Section 7.3 we show that, for any s > 1, the boundary ∂Ω can have positive Lebesgue measure. However, with a proper function φ, a dimension estimate for the boundary can be established by applying generalized mean porosity. It is indeed immediate that the boundary of a φ-John domain is weakly mean porous with parameters α(t) = Cφ(t) (for small t) and λ(k) ≥ 2^{−k}/2α(2^{−k}) (take 𝒬 to be the collection of the Whitney decompositions of all the cubes in the Whitney decomposition of R^{n} \ ∂Ω). By applying Corollary 3.5 we obtain the following result.

Corollary 6.2. Let Ω ⊂ R^{n} be a φ-John domain. Assume that

∫_{0} \frac{φ(t)^{n−1}dt}{t^{n}} = ∞.

Then m_{n}(∂Ω) = 0 and there is a positive constant C(n) and an integer j_{0} such that H^{h}(∂Ω) < ∞ for each premeasure h which satisfies

h(2^{−j}) ≤ M2^{−jn} exp(C(n) ∫_{[2^{−j},2^{−j_{0}} ]} \frac{φ(t)^{n−1} dt}{t^{n}})

for all j > j_{0} with some positive constant M.

Note that this corollary is essentially sharp by an example given in Section 7.3. Therefore, if

φ(t) = \frac{t}{(log(1/t))^{s}}

for small t, then we obtain a dimension bound when s ≤ 1/(n − 1), whereas the boundary can have positive volume when s > 1/(n − 1).

Remark 6.3. Let Ω ⊂ R^{n} be a φ-John domain with

φ(t) = t(log \frac{1}{t})^{−1/(n−1)}

for small t. Then Corollary 6.2 implies that m_{n}(∂Ω) = 0 and, moreover, H^{h}(∂Ω) < ∞ for the gauge function

h(t) = t^{n}(log \frac{1}{t})^{C(n)}.

7. Sharpness of the results

Recall the well-known Frostman lemma. We denote by ℳ (A) the set of Radon measures μ such that spt(μ) ⊂ A and μ(R^{n}) = 1.

Lemma 7.1. Let A be a Borel set in R^{n} and suppose there exists μ ∈ ℳ (A) such that μ(B(x, r)) ≤ h(r) for x ∈ R^{n} and r > 0. Then H^{h}(A) > 0.

Proof. Take any collection of balls B_{i} = B(x_{i}, r_{i}), x_{i} ∈ R^{n}, r_{i} < r, such that A ⊂ ⋃_{i} B_{i}. Then 1 = μ(A) ≤ μ(⋃_{i} B_{i}) ≤ ∑_{i} μ(B_{i}) ≤ ∑_{i} h(r_{i}).

7.1. Sharpness of Corollary 3.5. We show the sharpness of Corollary 3.5 by constructing an example of a set E which is weakly mean porous with parameters α and λ and for which H^{h}(E) > 0 with a premeasure h satisfying for all j ∈ Z^{+}

h(2^{−j}) ≤ 2^{−jn} exp(C̃ (n) ∑_{k=1}^{j} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} )

with some constant C̃ (n).

Let α and λ be as in Corollary 3.5. Then

(30)  p(k) = \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} is a decreasing function of k.

Let

Q_{0} = \{x ∈ R^{n} : |x_{i}| ≤ \frac{1}{4\sqrt{n}} for all i = 1, 2, …, n\}.

We define for each k ∈ Z^{+} a collection ℰ_{k} of closed sets F ⊂ R^{n} in the following way. ℰ_{0} consists of Q_{0} alone. To obtain ℰ_{k} from ℰ_{k−1} , subdivide Q_{0} into 2^{nk} closed dyadic cubes Q_{k}^{i} of diameter 2^{−k−1} in the natural way. For each i = 1, …, 2^{nk}, let Q̃ _{k}^{i} ⊂ Q_{k}^{i} be the smallest open cube in the center of Q_{k}^{i} such that it contains 2^{n}λ(k) disjoint open cubes of side length α(2^{−k}). We may assume that the diameter of Q̃ _{k}^{i} is dyadic and, by condition (30),

(31)  \frac{d(Q̃ _{k}^{i})}{d(Q_{k}^{i})} ≤ \frac{1}{4} for all i and k.

Let

ℰ_{k} = {F ∩ Q_{k}^{i} \ Q̃ _{k}^{i} : F ∈ ℰ_{k−1}, i = 1, …, 2^{nk}}.

We show that the set

E = ⋂_{k=1}^{∞} ⋃_{F ∈ℰ_{k}} F

is weakly mean porous with parameters α and λ.

Define for every k ∈ Z^{+} a collection 𝒬̃ _{k} of disjoint cubes by subdividing each Q̃ _{k}^{i}, i = 1, …, 2^{nk}, into 2^{n}λ(k) subcubes of side length α(2^{−k}). Then let

𝒬_{k} = {Q ∈ 𝒬̃ _{k} : Q ⊂ ⋃_{i=1}^{2^{nk}} Q̃ _{k}^{i} \ ⋃_{j=0}^{k−1} ⋃_{l=1}^{2^{nj}} Q̃ _{j}^{l}},

and let

𝒬 = {Q ∈ 𝒬_{k} : k = 1, 2, …}.

To see that 𝒬 indeed satisfies the conditions of Definition 3.1, notice the following geometric facts. Fix x ∈ E and take any Q_{k} (a dyadic subcube of Q_{0} with diameter 2^{−k−1} ) such that x ∈ Q_{k}. Now, there exists a dyadic subcube Q_{k+3} ⊂ Q_{k} such that Q_{k+3} ⊂ A_{k+3}(x). Moreover, by inequality (31), we can choose the cube Q_{k+3} so that it is not completely covered by the set ⋃_{j=0}^{k+2} ⋃_{l=1}^{2^{nj}} Q̃ _{j}^{l} (here Q̃ _{j}^{l} is defined as above). Now, by inequality (31), there is at most one dyadic subcube Q_{j} such that j ≤ k + 2 and Q̃_{j} ∩ Q̃_{k} ≠ ∅. Since the diameter of the cube Q̃_{j} is dyadic, we see that the set Q̃_{k+3} \ ⋃_{j=0}^{k+2} ⋃_{l=1}^{2^{nj}} Q̃ _{j}^{l} contains at least λ(k + 3) cubes Q ∈ 𝒬 such that l(Q) ≥ α(2^{−k−3}). Thus we have that

χ_{k}^{𝒬} (x) = 1

for every k and hence E is weakly mean porous.

Next we estimate the dimension of E by using Frostman’s lemma. Define the density

Δ_{k,F} = \frac{|⋃_{G∈ℰ_{k+1}} G ∩ F |}{|F |} = ∑_{G∈ℰ_{k+1}} \frac{|G ∩ F |}{|F |}

for each (nonempty) set F ∈ ℰ_{k}. The construction above implies

(32)  Δ_{k,F} ≥ (1 − \frac{C(n)λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} ) =: Δ_{k}.

For each set F ∈ ℰ_{k} let F_{i} ∈ ℰ_{i}, i = 0, 1, …, k − 1, be the (unique) sets for which F ⊂ F_{i}. We define a sequence of Radon measures μ_{k}, k = 0, 1, 2, …, so that

μ_{0}(A) = \frac{1}{|Q_{0}|}|Q_{0} ∩ A|

and

μ_{k}(A) = \frac{1}{|Q_{0}|} ∑_{F ∈ℰ_{k}} \frac{|F ∩ A|}{Δ_{k−1,F_{k−1}} × ⋯ × Δ_{0,F_{0}}}

for all measurable A ⊂ R^{n}. Then spt(μ_{k}) ⊂ ⋃_{F ∈ℰ_{k}} F and

\begin{aligned}μ_{k+1}(F) & = \frac{1}{|Q_{0}|} ∑_{G∈ℰ_{k+1}} \frac{|G ∩ F |}{Δ_{k,F} × Δ_{k−1,F_{k−1}} × ⋯ × Δ_{0,F_{0}}} \\ & = \frac{1}{|Q_{0}|} \frac{|F |}{Δ_{k−1,F_{k−1}} × ⋯ × Δ_{0,F_{0}}} = μ_{k}(F) \end{aligned}

for all F ∈ ℰ_{k}. Hence μ_{l}(F) = μ_{k}(F) for all l ≥ k. In particular, for all k

μ_{k}(R^{n}) = μ_{k}(Q_{0}) = μ_{0}(Q_{0}) = 1

and thus μ_{k} ∈ ℳ (R^{n}) for all k.

Recall that ℳ (R^{n}) is a compact metric space with an appropriate metric d (see [M, pp. 52–55]). Hence there is a subsequence μ_{k_{i}} converging to a measure μ ∈ ℳ (R^{n}) in the metric d. Note that spt(μ) ⊂ E.

Let j ∈ Z^{+} and let x ∈ R^{n}. By (32) we have for each G ∈ ℰ_{j+1} that

μ(G) ≤ \frac{|G|}{|Q_{0}| ∏_{k=1}^{j} Δ_{k}}

and, by the construction, the ball B(x, 2^{−j}) intersects at most P (n) sets G^{i} ∈ ℰ_{j+1} , where P (n) is a constant depending only on n. Hence

μ(B(x, 2^{−j})) ≤ μ( ⋃_{i=1}^{P (n)} G^{i}) ≤ \frac{∑_{i=i}^{P (n)} |G^{i}|}{|Q_{0}| ∏_{k=1}^{j} Δ_{k}} ≤ \frac{P (n)2^{−jn}}{|Q_{0}| ∏_{k=1}^{j} Δ_{k}} .

We choose the constant Q(n) ≥ P (n)/|Q_{0}| and the function

h(2^{−j}) = \frac{Q(n)2^{−jn}}{∏_{k=1}^{j} Δ_{k}} = Q(n)2^{−jn} exp(− log(∏_{k=1}^{j} Δ_{k}))

= Q(n)2^{−jn} exp(− ∑_{k=1}^{j} log Δ_{k})

= Q(n)2^{−jn} exp(− ∑_{k=1}^{j} log(1 − \frac{C(n)λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} )).

Clearly we have that

h(2^{−j}) ≤ 2^{−jn} exp(C̃ (n) ∑_{k=1}^{j} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} ),

when we choose the constant C̃ (n) large enough. Therefore

μ(B(x, 2^{−j})) ≤ h(2^{−j}) ≤ 2^{−jn} exp(C̃(n) ∑_{k=1}^{j} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} )

and the claim follows by Frostman’s lemma.

Note especially that if

∑_{k=1}^{∞} \frac{λ(k)α(2^{−k})^{n}}{(2^{−k})^{n}} < ∞,

then Frostman’s lemma implies that m_{n}(E) > 0.

7.2. Sharpness of Corollary 4.9. We show the sharpness of Corollary 4.9 in the case n = 2 by constructing a domain Ω ⊂ R^{2} such that it satisfies the quasihyperbolic growth condition with a function ϕ, and H^{h}(∂Ω) > 0 with a premeasure h satisfying

h(2^{−j}) ≤ 2^{−2j} exp(C̃ ∫_{[2^{−j},2^{−j_{0}} ]} \frac{dt}{−ϕ′(t)t^{2}} )

for all large j ∈ Z^{+} , where the constant C̃ depends only on β.

Let ϕ: ]0, 1] →]0, ∞[ be a function satisfying the conditions of Corollary 4.9. By condition (10) we can take j_{0} to be the smallest integer such that

\frac{4β^{2}}{−ϕ′(2^{−j_{0}})} ≤ \frac{2^{−j_{0}}}{16} .

Then define a function α so that it satisfies

α(2^{−k}) = \frac{4β^{2}}{−ϕ′(2^{−k})}

for all k ≥ j_{0} . Thus α(2^{−k})/2^{−k} ≤ 1/16 for all k ≥ j_{0} . We may assume that α(2^{−k}) is dyadic for all k ≥ j_{0} .

Let

Q_{j_{0}} = {(x, y) ∈ R^{2} : |x| ≤ 2^{−j_{0}−1} and |y| ≤ 2^{−j_{0}−1}}.

Let Ω_{j_{0}}^{x} be the α(2^{−j_{0}}) neighborhood of the x-coordinate axis in the square Q_{j_{0}} . Let Ω_{j_{0}}^{y} be the α(2^{−j_{0}}) neighborhood of the y -coordinate axis in the square Q_{j_{0}} . Let Ω_{j_{0}} = Ω_{j_{0}}^{x} ∪ Ω_{j_{0}}^{y}. For each k > j_{0} define Ω_{k} by subdividing Q_{j_{0}} into dyadic squares Q_{k}^{i}, i = 1, 2, …, 2^{2(k−j_{0})} , of side length 2^{−k}. Let Ω̃ _{k}^{x} be the union of the α(2^{−k}) neighborhoods of the centered x-coordinate axes in the squares Q_{k}^{i} and let Ω_{k}^{x} = Ω̃ _{k}^{x} \ ⋃_{i=j_{0}}^{k−1} \overline{Ω}_{i}. Similarly let Ω̃ _{k}^{y} be the union of the α(2^{−k}) neighborhoods of the centered y -coordinate axes in the squares Q_{k}^{i} and let Ω_{k}^{x} = Ω̃ _{k}^{x} \ ⋃_{i=j_{0}}^{k−1} \overline{Ω}_{i}. Let Ω_{k} = Ω_{k}^{x} ∪ Ω_{k}^{y}.

Define the domain Ω by

Ω = ⋃_{k=j_{0}}^{∞} Ω_{k} ∪ ⋃_{k=j_{0}}^{∞} int_{1}( \overline{Ω}_{k}^{x} ∩ \overline{Ω}_{k+1}^{y}),

where we denote by int_{1}( ·) the one-dimensional interior of the set.

Notice that the domain Ω satisfies the quasihyperbolic growth condition with the function ϕ: Let x_{0} = 0 and let x_{j} ∈ Ω such that α(2^{−j−1}) ≤ d(x_{j}, ∂Ω) ≤ α(2^{−j}) for some j ≥ 2j_{0} . Now we have that

k_{Ω}(x_{0}, x_{j}) ≤ 2 ∑_{k=j_{0}}^{j+1} ∫_{0}^{2^{−k}} \frac{dt}{α(2^{−k})} ≤ 2 ∑_{k=j_{0}}^{j+1} \frac{2^{−k}}{α(2^{−k})}

= \frac{1}{2β^{2}} ∑_{k=j_{0}}^{j+1} −ϕ′(2^{−k})2^{−k} ≤ \frac{1}{β^{2}} ∫_{2^{−j−1}}^{2^{−j_{0}+1}} −ϕ′(t) dt

≤ \frac{1}{β^{2}} ϕ(2^{−j−1}) ≤ ϕ(2^{−j/2}) ≤ ϕ(\frac{2^{−j}}{2^{−j_{0}}} )

≤ ϕ(\frac{α(2^{−j})}{α(2^{−j_{0}})}) ≤ ϕ(\frac{d(x_{j}, ∂Ω)}{d(x_{0}, ∂Ω)}).

Hence there is a constant C_{0} such that the inequality

k(x, x_{0}) ≤ ϕ(\frac{d(x, ∂Ω)}{d(x_{0}, ∂Ω)}) + C_{0}

holds for all x ∈ Ω.

We obtain the desired dimension estimate for ∂Ω similarly to Section 7.1. Let ℰ_{j_{0}} = {Q_{j_{0}} \ Ω_{j_{0}}} and define for all k > j_{0}

ℰ_{k} = {F ∩ Q_{k}^{i} \ Ω_{k} : F ∈ ℰ_{k−1}, i = 1, 2, …, 2^{2(k−j_{0})}}.

Now

∂Ω = E = ⋂_{k=j_{0}}^{∞} ⋃_{F ∈ℰ_{k}} F,

and for the density Δ_{k,F} we have the estimate

Δ_{k,F} ≥ (1 − \frac{Cα(2^{−k})}{2^{−k}})

for each F ∈ ℰ_{k}. Hence

Δ_{k,F} ≥ (1 − \frac{C}{−ϕ′(2^{−k})2^{−k}} ),

where the constant C depends only on β. The claim follows as in Section 7.1 by using Frostman’s lemma. Note especially that if the sum ∑ 1/−ϕ′(2^{−k})2^{−k} converges, or equivalently, if

∫_{0} \frac{dt}{−ϕ′(t)t^{2}} < ∞,

then m_{n}(∂Ω) > 0.

Note that this example also shows the essential sharpness of Corollary 5.2. Indeed, the domain Ω is simply connected and hence it is the image of the disk B^{2} for some quasiconformal mapping f: B^{2} → R^{2} . Since Ω satisfies the quasihyperbolic growth condition with the function ϕ which satisfies the conditions of Corollary 4.9, the mapping f is uniformly continuous with a corresponding modulus of continuity ψ(t) = Cϕ^{−1}(C log(1/t)); see [HK] for details. In this case the dimension estimates of the Corollaries 4.9 and 5.2 are essentially equivalent (except perhaps with different constants).

To prove the sharpness of the dimension estimate in R^{n} with n > 2, a similar construction can be carried out. We sketch an outline for the case n = 3. Let

Q_{j_{0}} = {x ∈ R^{3} : |x_{i}| ≤ 2^{−j_{0}−1} for all i = 1, 2, 3}.

Define Ω_{j_{0}} now by removing the α(2^{−j_{0}}) neighborhoods of the coordinate axes in Q_{j_{0}} and of the lines (t, ±2^{j_{0}−2}, 0) in Q_{j_{0}} . Define Ω_{j} accordingly for j > j_{0} in the dyadic subcubes of Q_{j_{0}} and, finally, define the domain Ω by attaching the x_{3} -components of Ω_{j+1} to the x_{1} -components of Ω_{j}. By [V] one can deduce that with this construction Ω is a quasiconformal ball.

7.3. Sharpness of Corollary 6.2. We show the sharpness of Corollary 6.2 in the case n = 2 by constructing a φ-John domain Ω ⊂ R^{2} for which H^{h}(∂Ω) > 0 with a premeasure h satisfying

h(2^{−j}) ≤ 2^{−2j} exp(C̃ ∫_{[2^{−j},2^{−j_{0}} ]} \frac{φ(t)dt}{t^{2}})

for all large j with some constant C̃ .

Let φ: ]0, 1[→]0, 1[ be a continuous function such that φ(t)/t is an increasing function. Choose α(t) = φ(t) and construct a domain Ω similarly as in Section 7.2. Notice that Ω is a φ-John domain: Let x_{0} = 0 be the John center, and for any 172 Tomi Nieminen

x ∈ Ω let γ(x_{0}, x) be the quasihyperbolic geodesic joining x to x_{0} . Now the length of γ is at most 2^{−j_{0}} and d(γ(t), ∂Ω) ≥ φ(t) for all 0 < t < l ≤ 2^{−j_{0}} .

We obtain the desired dimension estimate similarly as in Section 7.2. Now, for the density Δ_{k,F} , we have the following estimate. For each F ∈ ℰ_{k}

Δ_{k,F} ≥ (1 − \frac{Cφ(2^{−k})}{2^{−k}}).

The claim follows as in Section 7.1 by using Frostman’s lemma. To prove the sharpness of the dimension estimate in R^{n} with n > 2, a similar construction can be carried out. The case n = 3 is outlined at the end of Section 7.2.

References

[Bo] Bojarski, B.: Remarks on Sobolev Imbedding Inequalities. - In: Lecture Notes in Math. 1351, Springer-Verlag, Berlin, 1989, 52–68.

[F] Federer, F.: Geometric Measure Theory. - Springer-Verlag, Berlin–Heidelberg, 1969.

[G] de Guzmán, M.: Differentiation of Integrals in R^{n}. - Springer-Verlag, Berlin–Heidelberg, 1975.

[GO] Gehring, F. W., and B. Osgood: Uniform domains and the quasihyperbolic metric. - J. Analyse Math. 36, 1979, 50–74.

[HK] Hencl, S., and P. Koskela: Quasihyperbolic boundary conditions and capacity: Uniform continuity of quasiconformal mappings. - J. Anal. Math. (to appear).

[JM] Jones, P. W., and N. Makarov: Density properties of harmonic measure. - Ann. Math. 142, 1995, 427–455.

[KOT] Koskela, P., J. Onninen, and J. T. Tyson: Quasihyperbolic boundary conditions and capacity: Hölder continuity of quasiconformal mappings. - Comment. Math. Helv. 76, 2001 416–435.

[KR] Koskela, P., and S. Rohde: Hausdorff dimension and mean porosity. - Math. Ann. 309, 1997, 593–609.

[M] Mañe, R.: Ergodic Theory and Differentiable Dynamics. - Springer-Verlag, Berlin, 1987.

[MV] Martio, O., and M. Vuorinen: Whitney cubes, p-capacity, and Minkowski content. - Exposition. Math. 5, 1987, 17–40.

[R] Rogers, C. A.: Hausdorff Measures. - Cambridge Univ. Press, 1970.

[S] Stein, E. M.: Singular Integrals and Differentiability Properties of Functions. - Princeton Univ. Press, Princeton, 1970.

[SS1] Smith, W., and D. A. Stegenga: Hölder domains and Poincaré domains. - Trans. Amer. Math. Soc. 319, 1990, 67–100.

[SS2] Smith, W., and D. A. Stegenga: Exponential integrability of the quasiconformal metric in Hölder domains. - Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 1991, 345–360.

[T] Trotsenko, D. A.: Properties of regions with nonsmooth boundary. - Sibirsk. Mat. Zh. 22, 1981, 221–224 (Russian).

[V] Väisälä, J.: Quasiconformal maps and positive boundary measure. - Analysis 9, 1989, 205–216.

[W] Wloka, J.: Partial Differential Equations. - Cambridge Univ. Press, Cambridge, 1987.

Received 13 December 2004