On the $(1/2,+)$-caloric capacity of Cantor sets

TL;DR

This paper characterizes the $(1/2,+)$-caloric capacity of Cantor sets and relates BMO and Lipschitz variants to Hausdorff contents, extending capacity theory to fractional heat equations.

Contribution

It provides a detailed characterization of the $(1/2,+)$-caloric capacity for Cantor sets and connects BMO and Lipschitz variants to Hausdorff measures, advancing fractional capacity analysis.

Findings

Characterization of $(1/2,+)$-caloric capacity for Cantor sets.

Equivalence of BMO and Lipschitz variants with Hausdorff contents.

Analogies between fractional and classical Newtonian capacities.

Abstract

In the present paper we characterize the $(1/2,+)$-caloric capacity (associated with the $1/2$-fractional heat equation) of the usual corner-like Cantor set of $\mathbb{R}^{n+1}$. The results obtained for the latter are analogous to those found for Newtonian capacity. Moreover, we also characterize the BMO and $\text{Lip}_\alpha$ variants ($0<\alpha<1$) of the $1/2$-caloric capacity in terms of the Hausdorff contents $\mathcal{H}^n_\infty$ and $\mathcal{H}^{n+\alpha}_\infty$ respectively.