  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.
