Sparse non-smooth atomic decomposition of quasi-Banach lattices

A theory of non-smooth atomic decomposition is obtained fora large class of quasi-Banach lattices, including Morrey spaces, Lorentzspaces, mixed Lebesgue spaces as well as some related function spaces.As an application, an inequality comparing the fractional maximal operatorand the fractional integral operator is considered. Some examplesshow that the restriction posed on quasi-Banach lattices are indispensable.This paper, which is a follow-up of the third author’s paper in2020, simplifies the proof of some existing results