Gradient self-improvement for mixed local and nonlocal parabolic equations

Abstract

We introduce a new method for studying gradient higher integrability for mixed local and nonlocal parabolic equations. More precisely, for $p>2d/(d+2)$ and $s \in (0,1)$, we prove that if the inhomogeneity $F \in L^{p(1+\sigma)}_{\mathrm{loc}}$ for some $\sigma>0$, then every weak solution satisfies $\nabla u \in L^{p(1+\eps)}_{\mathrm{loc}}$ for some $\eps>0$, together with quantitative local estimates. The proof relies on stopping time arguments and an intrinsic Calder\'{o}n-Zygmund-type covering decomposition in which the inhomogeneity and nonlocal energy contributions are treated by classical and fractional maximal estimates.