[Paper Review] The Riesz Capacity in Metric Spaces

Juho Nuutinen, Pilar Silvestre|arXiv (Cornell University)|Jan 23, 2015

Advanced Harmonic Analysis Research21 references3 citations

TL;DR

This paper establishes a theory of Riesz capacity in metric spaces equipped with a doubling measure, defining a metric Riesz potential and capacity that generalize classical Euclidean capacity theory. It proves key properties like monotonicity, countable subadditivity, and Fatou's lemma-type lower semicontinuity, and provides sharp upper and lower bounds for the capacity in terms of a modified Hausdorff content, valid without Ahlfors regularity assumptions.

ABSTRACT

We study a capacity theory based on a definition of a Riesz potential in metric spaces with a doubling measure. In this general setting, we study the basic properties of the Riesz capacity, including monotonicity, countable subadditivity and several convergence results. We define a modified version of the Hausdorff measure and provide lower bound and upper bound estimates for the capacity in terms of the modified Hausdorff content.

Motivation & Objective

• To develop a capacity theory based on a metric Riesz potential in general metric measure spaces with doubling measures.
• To establish fundamental properties of the Riesz capacity, including monotonicity, countable subadditivity, and Fatou-type lower semicontinuity.
• To define a modified Hausdorff content and derive upper and lower bounds for the Riesz capacity in terms of this content.
• To show that the Riesz capacity is a Choquet capacity and to prove its equivalence to positive Hausdorff content under certain conditions.
• To extend classical Riesz capacity theory from Euclidean spaces to general metric measure spaces without requiring Ahlfors regularity.

Proposed method

• Define the metric Riesz potential of order $\gamma \in (0,1)$ as $I_\gamma f(x) = \int_X \frac{f(y)}{\mu(B(x,d(x,y)))^{1-\gamma}} \, d\mu(y)$, using only the measure of balls in the kernel.
• Use the doubling property of $\mu$ to derive measure growth estimates for balls, including $\mu(B(y,r))/\mu(B(x,R)) \geq C(r/R)^Q$ for $r \leq R$.
• Introduce a modified Hausdorff content $\widetilde{\mathcal{H}}_{\infty}^{\gamma,p}$ to estimate the Riesz capacity from above and below.
• Apply Hölder's inequality and a dyadic decomposition of balls to relate $L^p$ norms of test functions to capacity estimates.
• Use the $5r$-covering theorem to select disjoint balls covering the set $E$, enabling measure comparison via doubling.
• Establish equivalence between positive capacity and positive modified Hausdorff content under the condition $\tilde{\gamma}\tilde{p} < \gamma p < 1$.

Experimental results

Research questions

• RQ1 How can the classical Riesz capacity theory be extended to general metric measure spaces with doubling measures?
• RQ2 What are the fundamental properties (monotonicity, subadditivity, lower semicontinuity) of the metric Riesz capacity?
• RQ3 Can the Riesz capacity be bounded above and below using a modified Hausdorff content without assuming Ahlfors regularity?
• RQ4 Under what conditions does positive Riesz capacity imply positive modified Hausdorff content, and vice versa?
• RQ5 How does the capacity behave under approximation by compact and open sets, and is it a Choquet capacity?

Key findings

• The Riesz capacity $\mathcal{C}_{\gamma,p}$ is a Choquet capacity, meaning it can be approximated from inside by compact sets and from outside by open sets.
• The Riesz capacity satisfies Fatou-type lower semicontinuity, ensuring that $\liminf_{n\to\infty} \mathcal{C}_{\gamma,p}(E_n) \geq \mathcal{C}_{\gamma,p}(E)$ for decreasing sequences of sets.
• For any Borel set $E$, $\mathcal{C}_{\gamma,p}(E) > 0$ implies $\widetilde{\mathcal{H}}^{\gamma,p}(E) > 0$ under the condition $\gamma p < 1$.
• If the space is connected and $\tilde{\gamma}\tilde{p} < \gamma p < 1$, then $\widetilde{\mathcal{H}}^{\tilde{\gamma},\tilde{p}}(E) > 0$ implies $\mathcal{C}_{\gamma,p}(E) > 0$.
• The capacity of a set $E$ is bounded above by a constant multiple of $\|f\|_{L^p}^p$ for any non-negative $f$ with $f \geq \chi_E$ and $\|f\|_{L^p}^p < \epsilon$, leading to $\mathcal{C}_{\gamma,p}(E) = 0$ as $\epsilon \to 0$.
• The modified Hausdorff content $\widetilde{\mathcal{H}}_{\infty}^{\gamma,p}$ provides both upper and lower bounds for the Riesz capacity, with constants depending only on doubling and dimension parameters.