Commutators of vector-valued intrinsic square functions on vector-valued generalized weighted Morrey spaces

TL;DR

This paper establishes boundedness properties for vector-valued intrinsic square functions and their commutators on generalized weighted Morrey spaces, expanding understanding of these operators in complex function spaces.

Contribution

It provides new boundedness results for vector-valued intrinsic square functions and their commutators on generalized weighted Morrey spaces without requiring monotonicity of the controlling functions.

Findings

Boundedness of intrinsic square functions on weighted Morrey spaces.

Boundedness of commutators of these functions on the same spaces.

Conditions for boundedness involve Zygmund-type integral inequalities.

Abstract

In this paper, we will obtain the strong type and weak type estimates for vector-valued analogues of intrinsic square functions in the generalized weighted Morrey spaces $M^{\Phi,\varphi}_{w}(\mathbb{R}^n)$. We study the boundedness of intrinsic square functions including the Lusin area integral, Littlewood-Paley $\mathrm{g}$-function and $\mathrm{g}_{\lambda}^{*}$ -function and their commutators on vector-valued generalized weighted Morrey spaces $M^{\Phi,\varphi}_{w}(l_2)$. In all the cases the conditions for the boundedness are given either in terms of Zygmund-type integral inequalities on $\varphi(x,r)$ without assuming any monotonicity property of $\varphi(x,r)$ on $r$.