Multilinear estimates for Calderón commutators

Abstract

In this paper, we investigate the multilinear boundedness properties of the higher ($n$-th) order Calderón 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 Calderón 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 Calderón 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 Calderón commutator with a rough homogeneous kernel.

arXiv comment33 pages, 1 figure, this text overlap with arXiv 1710.09664. International Mathematics Research Notices, to appear