Riesz potential estimates for mixed local-nonlocal problems with measure data

Abstract

We study gradient regularity for mixed local-nonlocal problems modelled upon \[ -\Delta_p u +(-\Delta_p)^su=\mu\qquad\text{for} \quad 2-\tfrac{1}{n}<p<\infty\quad \text{and}\quad s\in(0,1)\,,\] where $\mu$ is a bounded Borel measure. We prove pointwise bounds for the gradient $Du$ in terms of the truncated 1-Riesz potential of $\mu$.