Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

TL;DR

This paper establishes weighted inequalities for multilinear commutators of Marcinkiewicz integrals with bounded kernels, extending known results to weighted Lebesgue spaces and weak-type estimates.

Contribution

It proves new weighted inequalities for multilinear commutators of Marcinkiewicz integrals with bounded kernels, including weak-type estimates for the case p=1.

Findings

Weighted inequalities for al_{\u03a9,\u21b5} with _{} weights.

Boundedness of al_{\u03a9,} on weighted Lebesgue spaces.

Weak (L)^{1/r} estimate for p=1.

Abstract

Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following weighted inequalities are proved for $\omega\in A_{\infty}$ and $0<p<\infty$, $$\|\mu_{\Omega}(f)\|_{L^{p}(\omega)}\leq C\|M(f)\|_{L^{p}(\omega)}, \ \ \|\mu_{\Omega,\vec{b}}(f)\|_{L^{p}(\omega)}\leq C\|M_{L(\log L)^{1/r}}(f)\|_{L^{p}(\omega)}.$$ The weighted weak $L(\log L)^{1/r}$ -type estimate is also established when $p=1$ and $\omega\in A_{1}$.