Removable singularities for Lipschitz fractional caloric functions in time varying domains

TL;DR

This work extends removability theory for Lipschitz caloric functions to the fractional, time-varying setting by defining the Lipschitz $s$-caloric capacity $\Gamma_{\Theta^s}$ for the operator $\Theta^s=(-\Delta)^s+\partial_t$ with $s\in(1/2,1]$. It develops a localization framework for potentials tied to the kernels $\nabla_x P_s$ and $\partial_t^{1/(2s)} P_s$, proves that removability is equivalent to vanishing capacity, and identifies the critical dimension as $n+1$. The paper constructs Cantor-type $s$-parabolic sets with positive $\mathcal{H}^{n+1}_{p_s}$-measure that are removable, showing capacity vs. Hausdorff content diverges in general. It also demonstrates non-comparability between the parabolic Lipschitz capacity $\Gamma_{\Theta}$ and a related $\gamma^{1/2}_{\Theta}$ capacity in the plane, highlighting nuanced nonlocal-time effects. Together, these results generalize parabolic removability to the fractional regime and reveal intricate interactions between nonlocal diffusion in space and time.

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.

Figures (1)

• Figure 1: First iterates involved in the construction of $E_{p_1}$ and $E_{p_{2/3}}$ in $\mathbb{R}^3$. For $s=1$ we have chosen $\lambda_1 := 12^{-1/3}$, and for $s=2/3$ we have chosen $\lambda_1:=1/4$.

Theorems & Definitions (42)

• Theorem
• Theorem
• Theorem
• Theorem 2.1
• Remark 2.1
• Definition 2.1
• Definition 2.2
• Definition 3.1
• Definition 3.2
• Theorem 3.1