Gradient self-improvement for mixed local and nonlocal parabolic equations

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 $F \in L^{p(1+σ)}_{\mathrm{loc}}$ for some $σ>0$, then every weak solution satisfies $\nabla u