Boundedness of non-local operators with spatially dependent coefficients and $L_p$-estimates for non-local equations

Abstract

We prove the boundedness of the non-local operator \[ \mathcal{L}^a u(x)=\int_{\mathbb{R}^d} \left(u(x+y)-u(x)-χ_α(y)\big(\nabla u(x),y\big)\right) a(x,y)\frac{dy}{|y|^{d+α}} \] from $H_{p,w}^{\alpha }({\mathbb{R}}^d)$ to $L_{p,w}({\mathbb{R}}^d)$ for the whole range of $p\in (1,\infty )$, where $w$ is a Muckenhoupt weight. The coefficient $a(x,y)$ is bounded, merely measurable in $y$, and Hölder continuous in $x$ with an arbitrarily small exponent. We extend the previous results by removing the largeness assumption on $p$ as well as considering weighted spaces with Muckenhoupt weights. Using the boundedness result, we prove the unique solvability in $L_p$ spaces of the corresponding parabolic and elliptic non-local equations.