Removable Sets for Fractional Heat and Fractional Bessel-Heat Equations
Mouna Chegaar, Á. P. Horváth
TL;DR
The paper investigates removability of sets for fractional heat equations driven by the Laplace and Bessel-Laplace operators, formulating a capacity-based criterion in the $L^p_A$ framework. It develops a unified approach using a spherical modulus of smoothness and transform tools (Fourier for $a=0$, Hankel for $|a|>0$) to connect removability with vanishing capacities, via three dual functionals $N_{a,\\gamma,p'}^{1/p'}$, $Z_{a,\\gamma,p'}^{1/p'}$, and $C_{a,\\gamma,p'}^{1/p'}$ and the fundamental solution $P_{\\gamma,a}$. The results establish that zero $Z$-capacity implies removability, while removability implies zero $C$-capacity (with corresponding statements in the Laplace case $a=0$ and for $0<\\gamma<2$), thereby unifying the treatment of radial reductions and nonlocal diffusion in both settings. This framework extends known removability criteria from elliptic and Lipschitz contexts to time-fractional diffusion with exponents $\\gamma\in(0,2)$ and $p\in(1,\\infty)$, providing a versatile tool for analyzing singular sets via capacities in fractional diffusion problems.
Abstract
We examine the fractional heat diffusion equations $L_{γ,a}:=(-Δ_a)^{\fracγ{2}}+\partial_t$, where $Δ_a$ is the Laplace- or the Bessel-Laplace operator. We give conditions for removability which are sufficient and which are necessary, by $L^p$-capacities. Introducing a spherical modulus of smoothness we can treat the Laplace and Bessel-Laplace cases together.