Multilinear estimates for Calder\'on commutators

TL;DR

This paper establishes all endpoint multilinear boundedness estimates for higher-order Calderón commutators in higher dimensions, generalizing known one-dimensional results and exploring limitations for certain target spaces.

Contribution

It provides the first comprehensive endpoint estimates for multilinear Calderón commutators in dimensions greater than two, including new counterexamples and potential applications.

Findings

Established multilinear endpoint estimates for Calderón commutators in higher dimensions.

Proved the boundedness of the commutator from specific Lorentz and Lebesgue spaces.

Identified limitations of these estimates for certain target space exponents.

Abstract

In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calder\'on commutator for dimensions larger than two. We establish all multilinear endpoint estimates for the target space $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, including that Calder\'on commutator maps the product of Lorentz spaces $L^{d,1}(\mathbb{R}^d)\times\cdots\times L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{\frac{d}{d+n},\infty}(\mathbb{R}^d)$, which is the higher dimensional nontrivial generalization of the endpoint estimate that the $n$-th order Calder\'on commutator maps $L^{1}(\mathbb{R})\times\cdots\times L^{1}(\mathbb{R})\times L^1(\mathbb{R})$ to $L^{\frac{1}{1+n},\infty}(\mathbb{R})$. When considering the target space $L^{r}(\mathbb{R}^d)$ with $r<\frac{d}{d+n}$, some counterexamples are given to show that these multilinear estimates may not hold. The method in the present paper seems to have a wide range of applications and it can be applied to establish the similar results for Calder\'on commutator with a rough homogeneous kernel.