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 $p>2d\mathrm{/}(d+2)$ and $s\in (0,1)$, we prove that if the inhomogeneity $F\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{p(1+\sigma )}$ for some $\sigma >0$, then every weak solution satisfies $\mathrm{\nabla }u\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{p(1+\text{eps})}$ for some $\text{eps}>0$, together with quantitative local estimates. The proof relies on stopping time arguments and an intrinsic Calderón-Zygmund-type covering decomposition in which the inhomogeneity and nonlocal energy contributions are treated by classical and fractional maximal estimates.