Maximal operator for the higher order Calder\'on commutator

TL;DR

This paper studies the boundedness of a complex multilinear operator called the higher order Calderón commutator in higher dimensions, establishing new weighted estimates and endpoint results.

Contribution

It provides the first comprehensive weighted multilinear boundedness results for the higher order Calderón commutator in dimensions greater than two.

Findings

Established weighted multilinear estimates for the operator.

Derived endpoint estimates in higher dimensions.

Extended known results to more general weighted spaces.

Abstract

In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calder\'on commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calder\'on commutator.