Non-smooth decomposition of homogeneous Triebel-Lizorkin-Morrey spaces

The aim of this paper is to develop a theory of non-smooth decomposition in Triebel-Lizorkin-Morrey spaces. As a byproduct, we obtain the non-smooth decomposition results for Hardy spaces and Morrey spaces. The result extends what Frazier and Jawerth obtained in 1990 with the parameters subject to a condition. Unlike this foregoing work, the result in this paper is valid for all admissible parameters for Triebel-Lizorkin-Morrey spaces. As an application, an improvement of the Olsen inequality is obtained.