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 $$\begin{aligned} \mathcal {L}^a u(x)=\int _{\mathbb {R}^d} \left( u(x+y)-u(x)-\chi _\alpha (y)\big (\nabla u(x),y\big )\right) a(x,y)\frac{dy}{|y|^{d+\alpha }} \end{aligned}$$ 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.