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

초록

We prove the boundedness of the non-local operator L(a)u(x) = integral R-d ( u(x + y) - u(x) -x alpha( y)( del u(x), y)) a(x, y) dy/|y|(d+alpha) from H-p,w(alpha)(R-d) to L (p,w)(R-d) for the whole range of p is an element of(1,infinity), where w is aMuckenhoupt weight. The coefficient a( x, y) is bounded, merely measurable in y, and Holder 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.

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