Removable singularities for Lipschitz fractional caloric functions in time varying domains

TL;DR

This paper investigates the conditions under which singularities can be removed for Lipschitz solutions of the fractional heat equation, introducing a new capacity concept and analyzing associated singular integral operators.

Contribution

It introduces a Lipschitz fractional caloric capacity and studies its critical dimension and boundedness properties of related singular integral operators.

Findings

Defined a new Lipschitz fractional caloric capacity

Analyzed the critical dimension for removability of singularities

Established $L^2$-boundedness of key singular integral operators

Abstract

In this paper we study removable singularities for regular $(1,\frac{1}{2s})$-Lipschitz solutions of the $s$-fractional heat equation for $1/2<s<1$. To do so, we define a Lipschitz fractional caloric capacity and study its critical dimension and the $L^2$-boundedness of a pair of singular integral operators, whose kernels will be the gradient of the fundamental solution of the fractional heat equation and its conjugate.