Sparse Non-Smooth Atomic Decomposition of Quasi-Banach Lattices

A theory of non-smooth atomic decomposition is obtained for a large class of quasi-Banach lattices, including Morrey spaces, Lorentz spaces, mixed Lebesgue spaces as well as some related function spaces. As an application, an inequality comparing the fractional maximal operator and the fractional integral operator is considered. Some examples show that the restriction posed on quasi-Banach lattices are indispensable. This paper, which is a follow-up of the third author’s paper in 2020, simplifies the proof of some existing results.