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

We prove the boundedness of the non-local operator Lau(x)=∫Rd(u(x+y)-u(x)-χα(y)(∇u(x),y))a(x,y)dy|y|d+αfrom Hp,wα(Rd) to L_p,w(R^d) for the whole range of p∈ (1 , ∞) , 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.