On fractional parabolic $\text{BMO}$ and $\text{Lip}_{\alpha}$ caloric capacities

TL;DR

This paper characterizes removable sets for fractional heat equation solutions with parabolic BMO or Lip_alpha conditions by introducing fractional caloric capacities linked to parabolic Hausdorff content.

Contribution

It introduces fractional caloric capacities and establishes their equivalence to parabolic Hausdorff content for analyzing removability in fractional heat equations.

Findings

Fractional caloric capacities are comparable to parabolic Hausdorff content.

Removability of sets is characterized via these capacities.

New tools for fractional parabolic PDE analysis.

Abstract

In the present paper we characterize the removable sets for solutions of the fractional heat equation satisfying some parabolic $\text{BMO}$ or $\text{Lip}_\alpha$ normalization conditions. We do this by introducing associated fractional caloric capacities, that we show to be comparable to a certain parabolic Hausdorff content.