Boundedness of the Commutators on Triebel-Lizorkin Spaces and the Characterizations of Hajlasz – Sobolev Spaces

Abstract

We show the multilinear singular integral operators on Triebel- Lizorkin and Lebesgue spaces and on Function spaces. We characterize the new bases and variable with applications of Triebel-Lizorkin and Besov spaces. We establish the Sobolev spaces on arbitrary metric space, Lebesgue points and maximal functions. We consider an integral and new characterizations of Hajlasz–Sobolev space on metric spaces and fractals. We investigate the boundedness of a class of super and rough singular integral operators and the associated certain commutators on Triebel–Lizorkin spaces. We discuss the decompositions and characterizations of Besov–Hausdorff and Hajlasz-Sobolev and Triebel –Lizorkin–Hausdorff spaces and their applications with grand Littlewood–Paley functions.