Removability of product sets for Sobolev functions in the plane

TL;DR

This paper investigates how the geometric properties of sets in the plane influence their removability for Sobolev functions, revealing critical dimensions and regularity conditions that determine when product sets are removable.

Contribution

It establishes new criteria based on Hausdorff dimension and Ahlfors-regularity for the removability of product sets for Sobolev functions in the plane.

Findings

Hausdorff dimension of C determines a critical exponent for removability.

Removability depends on the size and structure of F, with thresholds based on dimension.

Ahlfors-regularity of C ensures removability for all totally disconnected F.

Abstract

We study conditions on closed sets $C,F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.