Lipschitz potential estimates for diffusion with jumps

For $p\in (1,\mathrm{\infty })$ and $s\in (0,1)$, we consider the following mixed local-nonlocal equation $$-{\mathrm{\Delta }}_{p}u+(-{\mathrm{\Delta }}_p{)}^{s}u=f\text{in}\mathrm{\Omega },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a bounded domain and the function $f\in L_{loc}^1(\mathrm{\Omega })$. Depending on the dimension $d$, we prove gradient potential estimates of weak solutions for the entire ranges of $p$ and $s$. As a byproduct, we recover the corresponding estimates in the purely diffusive setup, providing connections between the local and nonlocal aspects of the equation. Our results are new, even for the linear case $p=2$.