A Refined Notion of Perfectness

and Potential Theory

Toshiyuki Sugawa

Department of Mathematics, Graduate School of Science,

Hiroshima University, Higashi-Hiroshima, 739-8526 Japan

sugawa@math.sci.hiroshima-u.ac.jp
http://sugawa@cajpn.org/

July 28, 2004, ICU, Tokyo Setup and uniform perfectness.

Let (X, ρ) be a complete metric space. In what follows, otherwise stated, the space X is always assumed to be a complete separable metric space, although some results may be valid for more general spaces.

A non-empty closed subset E of X is called perfect if E has no isolated points, namely, each point a is an accumulation point of E\{a}.

Typical examples are continua or Cantor sets in the Euclidean space ℝ^{n}.

Uniform perfectness.

A non-empty closed subset E of X is called uniformly perfect if ∃r_{0} ∈ (0, +∞], ∃c ∈ (0, 1] s.t. E ∩ A(a, cr, r) ≠ ∅ for each a ∈ E and r ∈ (0, r_{0}), where

A(a, t, r) = {x ∈ X : t ≤ ρ(x, a) ≤ r}.

This notion first appeared in a paper by Beardon and Pommerenke in 1978 and was investigated more by Pommerenke later.

We will use the notation

B(a, r) = {x ∈ X : ρ(x, a) ≤ r}.

Characterizations.

Theorem 1 (Pommerenke, Järvi-Vuorinen) Let X = ℝ^{2} = ℂ and let E be a non-empty closed subset of X. TFAE.

1. E is uniformly perfect.

2. ∃r_{0} > 0, ∃c > 0 s.t.

Cap (E ∩ B(a, r)) ≥ cr for ∀r ∈ (0, r_{0}).

3. ∃r_{0} > 0, ∃α > 0, ∃c > 0 s.t.

H_{α}^{∞}(E ∩ B(a, r)) ≥ cr^{α} for ∀r ∈ (0, r_{0}).

Here Cap and H_{α}^{∞} mean the logarithmic capacity and α-dimensional Hausdorff content, respectively.

Note: This theorem is quantitative, i.e., the constants are estimated by each other. Some are generalized to the case X = ℝ^{n}, n ≥ 2.

A more refined notion of perfectness.

Suppose that a non-decreasing function φ : (0, r_{0}) → ℝ is given so that

0 < φ(r) ≤ r, 0 < ∀r < r_{0},

for some r_{0} ∈ (0, +∞]. We will say that a nonempty closed subset E of X is φ-perfect if E ∩ A_{φ}(a, r) ≠ ∅ for each a ∈ E and 0 < r < min {r_{0}, diam E/2}, where

A_{φ}(a, r) = {x ∈ X; φ(r) ≤ ρ(x, a) ≤ r}.

Note: φ-perfect sets are nothing but uniformly perfect sets when φ(r) = cr for some 0 < c ≤ 1.

Hausdorff contents and measures.

Let h be a gauge function, in other words, a strictly increasing, continuous, positive function on (0, +∞) with h(+0) = 0. We denote by H_{h} the Hausdorff h-measure and by H_{h}^{∞} the Hausdorff h-content.

More precisely, for a bounded Borel set E ⊂ X and for t ∈ (0, +∞], we set

H_{h}^{t}(E) = inf_{{E⊂∪jUj \atop diam Uj<t}} _{j}^{∑} h(diam U_{j}),
H_{h}(E) = lim_{t→0} H_{h}^{t}(E).

Note that H_{h}^{∞}(E) > 0 if and only if H_{h}(E) > 0.

When h(r) = r^{α}, we also write H_{h}^{t}(E) = H_{α}^{t}(E).

Generalized capacity.

This notion goes back to Frostman’s thèse (1935) in the case when (X, ρ) is a Euclidean space. Kametani (1945) treats the general case and deduce fundamental properties of generalized capacities.

Φ : (0, ∞) → ℝ : a capacity kernel, namely, a continuous, strictly decreasing function with Φ(+0) = +∞.

P (E) : the set of Borel probability measures μ on X with μ(E) = 1 for a Borel set E.

u_{μ}^{Φ} : the Φ-potential of μ ∈ P (X) given by

u_{μ}^{Φ} (x) = ∫_{X} Φ(ρ(x, y))dμ(y), x ∈ X.

Note: u_{μ}^{Φ} is lower semi-continuous on X.

Set

V^{Φ} (E) = inf_{μ∈P (E)} ∥u_{μ}^{Φ} ∥_{∞},

where

∥u_{μ}^{Φ} ∥_{∞} = sup_{x∈X} u_{μ}^{Φ} (x).

C^{Φ} (E) : the Φ-capacity of E defined by

C^{Φ} (E) = Φ^{−1}(V^{Φ} (E)).

Note that C^{Φ} (E) ≤ diam E.

When Φ(r) = − log r, Cap (E) = C^{Φ} (E) is called the logarithmic capacity of E. When Φ(r) = r^{−α}, C^{Φ} (E) is called the Newton capacity of order α > 0 (or (2 + α)-dimensional Newton capacity).

Relation between Hausdorff contents and generalized capacities.

The following generalizes a result of Erdös-Gillis.

Theorem 2 (Kametani (1945)) Suppose that X is a complete separable metric space. Let h be a gauge function and E be a compact subset of X. When Φ(r) = 1/h(r), the condition H_{h}(E) < ∞ implies C^{Φ} (E) = 0.

We also obtain the following weaker but quantitative result:

Lemma 3 Suppose that X is a complete metric space. Let h be a gauge function and E be a compact subset of X. If Φ(r) = 1/h(r),

\frac{1}{V^{Φ} (E)} ≤ H_{h}^{∞}(E).

An upper estimate for V^{Φ} (E) is also deduced in the following way when (X, ρ) is Euclidean.

Theorem 4 Assume that X = ℝ^{n}, ρ(x, y) = |x − y| and that

− ∫_{0}^{r_{1}} h(r)dΦ(r) < +∞

for some r_{1} > 0. For a compact subset E of X,

V^{Φ} (E) ≤ Φ(r_{0}) − \frac{A_{n}}{H_{h}^{∞}(E)} ∫_{0}^{r_{0}} h(t)dΦ(t)

holds, where r_{0} = 2diam E and A_{n} is a constant depending only on n.

Corollary 5 (Kametani) Under the same assumptions, H_{h}^{∞}(E) > 0 ⇒ C^{Φ} (E) > 0.

Main Theorem 1.

Let h : (0, r_{0}) → (0, +∞) be a gauge function. We define the functions ε_{j} by the relations

h(φ(x/3)) = \frac{exp ε_{1}(x)}{2} h(x)

and

h(2x) = \frac{exp ε_{2}(x)}{2} h(x)

for sufficiently small x.

For a function λ : (0, x_{0}) → ℝ, we say that a function ν : (0, x_{0}) → ℝ is a monotone majorant of λ if ν is increasing and satisfies |λ(x)| ≤ ν(x) for 0 < x < x_{0}.

Theorem 6 Let (X, ρ) be a complete metric space. Suppose that φ satisfies φ(r) ≤ cr in 0 < r < r_{0} for some constant 0 < c < 1. If there is a monotone majorant ω_{1} of ε_{1} such that ∫_{0}^{r_{0}} \frac{ω_{1}(r)dr}{r} < +∞, then

H_{h}^{∞}(E) ≥

\frac{h(δ_{0})}{2} exp (−ω_{1}(δ_{0}) − \frac{1}{log (6/c)} ∫_{0}^{δ_{0}} \frac{ω_{1}(x)dx}{x} )

holds for any φ-perfect subset E of X, where δ_{0} is an arbitrary number satisfying 0 < δ_{0} < min {r_{0}, diam E/2}. In particular, H_{h}^{∞}(E) > 0. In addition, if c < 1/4 and if there is a monotone majorant ω_{2} of ε_{2} such that ∫_{0}^{r_{0}} \frac{ω_{2}(r)dr}{r} < +∞, then for each d_{0} ∈ (0, r_{0}) there exists a φperfect compact set E ⊂ [0, d_{0}] such that

H_{h}(E) ≤

h(d_{0}) exp (ω_{2}(d_{0}) + \frac{1}{log (1/2c)} ∫_{0}^{d_{0}} \frac{ω_{2}(x)dx}{x} ) ,

and thus 0 < H_{h}^{∞}(E) ≤ H_{h}(E) < +∞.

Remark 1: When X is a Euclidean space, by Theorem 4, we may deduce a lower estimate of generalized capacities for φ-perfects sets.

Corollary 7 Let (X, ρ) be a complete metric space. If φ(r) = cr, 0 < r < r_{0} for some constants c ∈ (0, 1] and r_{0} > 0, then any φ-perfect set E satisfies H-dim E ≥ log 2/ log (6/c).

Remark 2: This fact was first shown by Järvi-Vuorinen (1996) when X = ℝ^{n} with a lower bound depending on the dimension n.

Remark 3: We cannot apply the above theorem to the case when φ(r) = ct^{β} for constants c > 0 and β > 0 because the Dini-type condition is not satisfied.

Main Theorem 2.

Theorem 8 Let (X, ρ) be a complete metric space. Suppose that φ satisfies φ(r) ≤ cr^{α} in 0 < r < r_{0} for some constants c > 0 and α > 1. If there is a monotone majorant ω_{1} of ε_{1} such that ∫_{0}^{r_{0}} \frac{ω_{1}(r)dr}{r log (2r_{0}/r)} < +∞, then

H_{h}^{∞}(E) ≥

\frac{h(δ_{0})}{2} exp (−ω_{1}(δ_{0}) − \frac{1}{log α} ∫_{0}^{δ_{0}} \frac{ω_{1}(x)dx}{x log (M/x)} )

holds for any φ-perfect subset E of X, where δ_{0} is an arbitrary number satisfying cδ_{0}^{α−1} ≤ 1 and 0 < δ_{0} < min {r_{0}, diam E} and

M = (\frac{2 · 3^{α}}{c} )^{1/(α−1)} (> δ_{0}).

In particular, H_{h}^{∞}(E) > 0.

Theorem 9 Under the same assumptions, if, in addition, there is a monotone majorant ω_{2} of ε_{2} such that ∫_{0}^{r_{0}} \frac{ω_{2}(r)dr}{r log (2r_{0}/r)} < +∞, then for each d_{0} ∈ (0, r_{0}) with cd_{0}^{α−1} < 1/4 there exists a φ-perfect compact set E ⊂ [0, d_{0}] such that

H_{h}(E) ≤

h(d_{0}) exp (ω_{2}(d_{0}) + \frac{1}{log α} ∫_{0}^{d_{0}} \frac{ω_{2}(x)dx}{x log (M/x)} )

and thus 0 < H_{h}^{∞}(E) ≤ H_{h}(E) < +∞.

An application.

Set φ(r) = cr^{α} (0 < r < r_{0}) for constants c > 0, α > 1, r_{0} > 0 with cr_{0}^{α} ≤ r_{0}. For this, we take h(t) = (log (2r_{0}/t))^{−γ}, 0 < t < r_{0}, where γ = \frac{log 2}{log α}. Then Theorems 8 and 9 are now applicable. Thus, as a corollary, we obtain the following.

Theorem 10 Let (X, ρ) be a complete metric space. Under the above assumptions, every φperfect compact set E ⊂ X satisfies H_{h}^{∞}(E) > 0. Furthermore, there is a compact set E ⊂ ℝ such that 0 < H_{h}^{∞}(E) ≤ H_{h}(E) < ∞.

By using Erdös-Gillis theorem, we have

Corollary 11 Under the same assumptions, every φ-perfect set has positive logarithmic capacity whenever 1 < α < 2. On the other hand, when α ≥ 2, there is a compact φ-perfect subset E of ℝ of logarithmic capacity zero.