Two-weighted estimates for some sublinear operators on generalized weighted Morrey Spaces and applications

Abstract: In this paper we investigate the boundedness of sublinear operators generated by fractional integrals as well as sublinear operators generated by Calder`on-Zygmund operators on generalized weighted Morrey spaces and generalized weighted mixed-Morrey spaces. In particular we are interested in the strong-type estimate for $1<p<\infty$ and the weak estimate for $p=1$. Under some assumptions, we prove that the operators and their commutator with a BMO function are bounded on those function spaces with different weights. The results then imply the boundedness of fractional integrals with Gaussian kernel bounds as well as with rough kernels, fractional maximal integrals with rough kernels, and sublinear operators with rough kernels generated by Calder`on-Zygmund operators. Using the results, we obtain some regularity properties of the solution of some partial differential equations.