Endpoint estimates for commutators of intrinsic square functions in the Morrey type spaces

TL;DR

This paper investigates the boundedness of commutators of intrinsic square functions with BMO functions in Morrey-type spaces, establishing endpoint estimates and weak-type inequalities relevant for harmonic analysis.

Contribution

It provides new endpoint estimates for these commutators in weighted and generalized Morrey spaces, extending prior boundedness results to endpoint cases.

Findings

Established weighted weak L log L-type estimates for commutators.

Proved endpoint estimates in weighted Morrey spaces with A_1 weights.

Extended results to generalized Morrey spaces with growth functions.

Abstract

In this paper, the boundedness properties of commutators generated by $b$ and intrinsic square functions in the endpoint case are discussed, where $b\in BMO(\mathbb R^n)$. We first establish the weighted weak $L\log L$-type estimates for these commutator operators. Furthermore, we will prove endpoint estimates of commutators generated by $BMO(\mathbb R^n)$functions and intrinsic square functions in the weighted Morrey spaces $L^{1,\kappa}(w)$ for $0<\kappa<1$ and $w\in A_1$, and in the generalized Morrey spaces $L^{1,\Theta}$, where $\Theta$ is a growth function on $(0,+\infty)$ satisfying the doubling condition.