Some remarks on the sparse dominations for commutators of multi(sub)linear operator

Zhidan Wang

doi:10.1186/s13660-021-02585-z

Abstract

We establish pointwise sparse dominations for the iterated commutators of multi(sub)linear operators satisfying the W_{q} condition. As consequences, we present some quantitative weighted estimates for the commutators. In addition, we also obtain the Fefferman–Stein inequality, the Coifman–Fefferman inequality, and the local decay estimates regarding the iterated commutators.

Highlights

Sparse domination is a relatively new tool to prove weighted estimates for singular integrals
Li [4] established sparse domination theorem for multilinear singular integral operators with the kernel satisfying the Lr-Hörmander condition
Our main results of this paper are as follows

Summary

Introduction

Sparse domination is a relatively new tool to prove weighted estimates for singular integrals. Li [4] established sparse domination theorem for multilinear singular integral operators with the kernel satisfying the Lr-Hörmander condition. Motivated by the above works, the purpose of this paper is to establish a sparse domination for the iterated commutators of multi(sub)linear operator with weaker hypotheses than [4, 5]. Motivated by Lerner and Ombrosi [7], we assume that T is an operator satisfying the following Wq property instead of assuming T is bounded from Lq × · · · × Lq → Lq/m,∞: there is a nonincreasing function ψT,q such that, for any fj ∈ Lq(Q) with j = 1, . Wm), and each wi is a nonnegative function on Rd. w is said to satisfy the following Ap/r condition if [w]Ap/r := sup

When r

It is easy to have that l

Now we can write

Rn m

Findings

The kernel