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

Abstract We study gradient regularity for mixed local–nonlocal problems modeled upon− Δ p u + ( − Δ p ) s u = μ $$ -a}}_{p}u+{(-a}}_{p s}u=\mu \end{equation for2 − 1 / n < p < ∞ $2-1/n<p< $ ands ∈ ( 0 , 1 ) $s\in (0,1)$ , where μ $\mu$ is a bounded Borel measure andn ⩾ 2 $n slant 2$ is the ambient dimension. We prove pointwise bounds for the gradientD u $Du$ in terms of the truncated 1‐Riesz potential of| μ mu |$ .