Pointwise regularity of solutions for fully fractional parabolic equations

TL;DR

This paper studies the detailed pointwise regularity of solutions to fully fractional parabolic equations, establishing new regularity results and providing simplified proofs using novel definitions and integral representations.

Contribution

It introduces new pointwise regularity results for fractional parabolic equations and simplifies proofs with innovative definitions and integral representations.

Findings

Established $C^{k+eta}$ regularity depending on $eta= ext{alpha}+2s$

Unified proof approach using novel pointwise function space definitions

Integral representation and directional averages are key tools

Abstract

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $(\partial_t -\Delta)^{s} u = f,$ where $s\in(0,1)$. We establish $C^{k+\alpha+2s}$ or $C^{k+\alpha+2s,\ln} (k\geq 0,\alpha\in[0,1))$ pointwise regularity according to $\alpha+2s\notin \mathbb{Z}$ or $\alpha+2s\in \mathbb{Z}$, which imply the classical local regularity directly. We provide a simplified and unified proof by introducing novel equivalent definitions for pointwise function spaces. Moreover, the equivalent integral representation and directional average for fractional heat kernel play an important role in our discussion.