Estimates for vector-valued intrinsic square functions and their commutators on certain weighted amalgam spaces

TL;DR

This paper introduces new weighted amalgam spaces and extends Wilson's estimates for vector-valued intrinsic square functions and their commutators to these spaces, providing new boundedness and endpoint results.

Contribution

It extends strong and weak-type estimates for vector-valued intrinsic square functions and commutators to weighted amalgam spaces, including endpoint weak L log L estimates.

Findings

Established boundedness of vector-valued intrinsic square functions on weighted amalgam spaces.

Proved mapping properties of vector-valued commutators with BMO functions on these spaces.

Derived weighted weak L log L estimates for commutators at the endpoint.

Abstract

In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we deal with the vector-valued intrinsic square functions, which are given by \begin{equation*} \mathcal S_\gamma(\vec{f})(x) = \Bigg(\sum_{j=1}^\infty \big|\mathcal S_\gamma(f_j)(x) \big|^2\Bigg)^{1/2}, \end{equation*} where $0<\gamma\leq1$ and \begin{equation*} \mathcal S_\gamma (f_j)(x) = \left(\iint_{\Gamma(x)} \Big[\sup_{\varphi\in{\mathcal C}_\gamma} \big|\varphi_t*f_j(y) \big|\Big]^2 \frac{dydt}{t^{n+1}}\right)^{1/2}, \quad j=1,2,\dots. \end{equation*} In his fundamental work, Wilson established strong-type and weak-type estimates for vector-valued intrinsic square functions on weighted Lebesgue spaces. The goal of this paper is to extend his results to these weighted amalgam spaces. Moreover, we define vector-valued analogues of commutators with $BMO(\mathbb R^n)$ functions, and obtain the mapping properties of vector-valued commutators on the weighted amalgam spaces as well. In the endpoint case, we also establish the weighted weak $L\log L$-type estimates for vector-valued commutators in the setting of weighted Lebesgue spaces.