On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space

TL;DR

This paper investigates the maximal Sobolev regularity of distributions supported on subsets of Euclidean space, characterizing it in terms of set capacity, Hausdorff dimension, and providing explicit examples and bounds for various types of sets.

Contribution

It offers a comprehensive classification of maximal Sobolev regularity supported by subsets of Euclidean space, refining previous results and including new bounds and explicit constructions.

Findings

Maximal regularity characterized by Hausdorff dimension for zero measure sets.

Full classification of regularity as a function of p and corresponding p-values.

New lower bounds on regularity for fat Cantor sets using capacity and norm estimates.

Abstract

This paper concerns the following question: given a subset $E$ of $\mathbb{R}^n$ with empty interior and an integrability parameter $1<p<\infty$, what is the maximal regularity $s\in\mathbb{R}$ for which there exists a non-zero distribution in the Bessel potential Sobolev space $H^{s,p}(\mathbb{R}^n)$ that is supported in $E$? For sets of zero Lebesgue measure we apply well-known results on set capacities from potential theory to characterise the maximal regularity in terms of the Hausdorff dimension of $E$, sharpening previous results. Furthermore, we provide a full classification of all possible maximal regularities, as functions of $p$, together with the sets of values of $p$ for which the maximal regularity is attained, and construct concrete examples for each case. Regarding sets with positive measure, for which the maximal regularity is non-negative, we present new lower bounds on the maximal Sobolev regularity supported by certain fat Cantor sets, which we obtain both by capacity-theoretic arguments, and by direct estimation of the Sobolev norms of characteristic functions. We collect several results characterising the regularity that can be achieved on certain special classes of sets, such as $d$-sets, boundaries of open sets, and Cartesian products, of relevance for applications in differential and integral equations.