Non-smooth decomposition of homogeneous Triebel-Lizorkin spaces with applications to the Marcinkiewicz integral (The deepening of function spaces and its environment)

Abstract

Thc aim of this paper is to develop a theory of non-smooth decomposition in homogeneous Triebel-Lizorkin spaces. As a byproduct, we can recover the decomposition results for Hardy spaces as a special case The result extends what Frazicr and Jawerth obtained in 1990. The result by Frazier and Jawcrth covcrs only thc limited range of the parameters but the result in this paper is valid for all admissible parameters for Triebel-Lizorkin spaces. As an application of the main results, we prove that the Marcinkiewicz operator is boundcd. What is new in this paper is to reconstruct sequence spaces other than classical lp spaces.