Pointwise regularity of solutions for fully fractional parabolic equations

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $({\mathrm{\partial }}_t-\mathrm{\Delta }{)}^{s}u=f,$ where $s\in (0,1)$. We establish $C^{k+\alpha +2s}$ or $C^{k+\alpha +2s,\ln }(k\ge 0,\alpha \in [0,1))$ pointwise regularity according to $\alpha +2s\mathrm{\notin }\mathbb{Z}$ 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.