Applications of Hardy Spaces Associated with Ball Quasi-Banach Function Spaces

• Published in: Resultate der Mathematik Vol. 75; no. 1
• Main Authors: Wang, Fan, Yang, Dachun, Yang, Sibei
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.03.2020
• Subjects: Mathematics; Mathematics and Statistics; 42B25; Ball quasi-Banach function space; pseudo-differential operator; Calderón–Zygmund operator; Primary 42B30; 42B20; function; 47G30; Secondary 42B35; atom; Hardy space
• ISSN: 1422-6383; 1420-9012
• DOI: 10.1007/s00025-019-1149-x

Let X be a ball quasi-Banach function space satisfying some minor assumptions. In this article, the authors establish the characterizations of H X ( R n ) , the Hardy space associated with X , via the Littlewood–Paley g -functions and g λ ∗ -functions. Moreover, the authors obtain the boundedness of Calderón–Zygmund operators on H X ( R n ) . For the local Hardy-type space h X ( R n ) associated with X , the authors also obtain the boundedness of S 1 , 0 0 ( R n ) pseudo-differential operators on h X ( R n ) via first establishing the atomic characterization of h X ( R n ) . Furthermore, the characterizations of h X ( R n ) by means of local molecules and local Littlewood–Paley functions are also given. The results obtained in this article have a wide range of generality and can be applied to the classical Hardy space, the weighted Hardy space, the Herz–Hardy space, the Lorentz–Hardy space, the Morrey–Hardy space, the variable Hardy space, the Orlicz-slice Hardy space and their local versions. Some special cases of these applications are even new and, particularly, in the case of the variable Hardy space, the g λ ∗ -function characterization obtained in this article improves the known results via widening the range of λ .