Weighted Estimates for Vector-Valued Intrinsic Square Functions and Commutators in the Morrey-Type Spaces

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with BMO(ℝn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$BMO(\mathbb {R}^{n})$\end{document} functions are discussed. We first show the weighted strong-type and weak-type estimates of vector-valued intrinsic square functions in the Morrey-type spaces. Then, we obtain weighted strong-type estimates of vector-valued analogues of commutators in Morrey-type spaces. In the endpoint case, we establish the weighted weak LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L\log L$\end{document}-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weighted endpoint estimates of these commutator operators in Morrey-type spaces. In particular, we can obtain strong-type and endpoint estimates of vector-valued intrinsic square functions and their commutators in the weighted Morrey spaces and the generalized Morrey spaces.