A characterization of two-weighted inequalities for singular operators and their commutators in generalized weighted Morrey spaces on spaces of homogeneous type

Abstract

In this paper we give a characterization of two-weighted inequalities for
singular operators and their commutators in generalized weighted Morrey
spaces on spaces of homogeneous type $\mathcal{M}_{\omega }^{p,\varphi }(X)$%
. We prove the boundedness of the Calder\'{o}n-Zygmund singular operators $T$
and its commutators $[b,T]$ from the spaces $\mathcal{M}_{\omega
_{1}^{\delta }}^{p,\varphi _{1}}(X)$ to the spaces $\mathcal{M}_{\omega
_{2}^{\delta }}^{p,\varphi _{2}}(X)$, where $1<p<\infty $, $0<\delta <1$ and
$(\omega _{1},\omega _{2})\in \widetilde{A}_{p}(X)$. Finally we give
generalized weighted Morrey a priori estimates as applications of our
results.