Weighted estimates for vector-valued intrinsic square functions and commutators in the Morrey type spaces

TL;DR

This paper investigates the boundedness of vector-valued intrinsic square functions and their commutators in Morrey type spaces, providing new weighted estimates and endpoint results in these function spaces.

Contribution

It introduces weighted strong and weak type estimates for vector-valued intrinsic square functions and their commutators in Morrey type spaces, including endpoint cases.

Findings

Established weighted strong and weak type estimates in Morrey spaces.

Proved endpoint estimates in weighted Lebesgue and Morrey spaces.

Extended results to vector-valued intrinsic square functions and commutators.

Abstract

In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with $BMO(\mathbb R^n)$ 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 $L\log L$-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.