On fractional parabolic $\text{BMO}$ and $\text{Lip}_α$ caloric capacities
Joan Hernández, Joan Mateu, Laura Prat
TL;DR
The paper develops a comprehensive potential-theoretic framework for the fractional parabolic heat equation $Θ^s=(-Δ)^s+∂_t$, introducing fractional caloric capacities that quantify removability under parabolic ${\rm BMO}$ and Lip$_α$ gradient conditions. It builds a robust technical backbone with detailed kernel estimates and growth controls for admissible functions, enabling precise links between capacity notions and parabolic Hausdorff contents ${\mathcal H}^{m}_{∞,p_s}$. The main contributions are sharp equivalences: ${Γ}_{Θ^s,*}(E)\approx{\mathcal H}^{n+1}_{∞,p_s}(E)$ for $α<2s-1$ and ${Γ}_{Θ^s,α}(E)\approx{\mathcal H}^{n+1+α}_{∞,p_s}(E)$, extended to generalized capacities ${γ}_{Θ^s,*}^{σ}$ and ${γ}_{Θ^s,α}^{σ}$ with exponents $n+2σ$ and $n+2σ+α$. These results generalize prior parabolic capacity theory to the fractional and nonlocal parabolic setting and provide exact removability criteria in terms of parabolic Hausdorff content, with potential applications in fractional PDE regularity and potential theory.
Abstract
In the present paper we characterize the removable sets for solutions of the fractional heat equation satisfying some parabolic $\text{BMO}$ or $\text{Lip}_α$ normalization conditions. We do this by introducing associated fractional caloric capacities, that we show to be comparable to a certain parabolic Hausdorff content.