Multilinear operators on Hardy spaces associated with ball quasi-Banach function spaces
Jian Tan
TL;DR
The article develops a unified framework for the boundedness of multilinear Calderón–Zygmund operators, their maximal operators, and multilinear pseudodifferential operators on Hardy-type spaces built over ball quasi-Banach function spaces. By imposing three structural conditions on the ambient space and employing atomic decompositions and kernel estimates, the authors establish mapping properties from products of generalized Hardy spaces into corresponding ball quasi-Banach targets, as well as into local Hardy spaces, under sharp range assumptions for the exponents. The results are then instantiated in six concrete settings—weighted, variable, Morrey, mixed-norm, Lorentz, and Orlicz spaces—delivering new, core boundedness statements for these operators in broad contexts. This general theory subsumes and extends numerous known results, offering a versatile toolkit for analyzing multilinear operators in diverse harmonic-analysis environments.
Abstract
This paper establishes that multilinear Calderón--Zygmund operators and their maximal operators are bounded on Hardy spaces associated with ball quasi-Banach function spaces. Moreover, we also obtain the boundedness of multilinear pseudo-differential operators on local Hardy spaces associated with ball quasi-Banach function spaces. Since these (local) Hardy type spaces encompass a wide range of classical (local) Hardy-type spaces including weighted (local) Hardy spaces, variable (local) Hardy space, (local) Hardy--Morrey space, mixed-norm (local) Hardy space, (local) Hardy--Lorentz space and (local) Hardy--Orlicz spaces, the results presented in this paper are highly general and essentially improve the existing results.