Boundedness of $\theta$-type Calder\'on--Zygmund operators and commutators in the generalized weighted Morrey spaces

Abstract

In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then we discuss the strong type and weak type estimates for a class of Calder\'on--Zygmund type operators $T_\theta$ in these new Morrey type spaces. Furthermore, the strong type estimate and endpoint estimate of commutators $[b,T_{\theta}]$ formed by $b$ and $T_{\theta}$ are established. Also we study related problems about two-weight, weak type inequalities for $T_{\theta}$ and $[b,T_{\theta}]$ in the Morrey type spaces and give partial results.