Generalized Carleson Measure Spaces and Their Applications

Chin-Cheng Lin,Kunchuan Wang

doi:10.1155/2012/879073

Abstract

We introduce the generalized Carleson measure spaces CMOrα,qthat extend BMO. Using Frazier and Jawerth'sφ-transform and sequence spaces, we show that, forα∈Rand0<p≤1, the duals of homogeneous Triebel-Lizorkin spacesḞpα,qfor1<q<∞and0<q≤1are CMO(q'/p)-(q'/q)-α,q'and CMOr-α+(n/p)-n,∞(for anyr∈R), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel-Lizorkin spaces.

Highlights

In 1972, Fefferman and Stein 1 proved that the dual of H1 is the BMO space
We show the boundedness of wavelet multipliers and paraproduct operators
Q and P always mean the dyadic cubes in Rn, and, for r > 0, we denote by rQ the cube concentric with Q whose each edge is r times as long

Summary

Introduction

In 1972, Fefferman and Stein 1 proved that the dual of H1 is the BMO space. In 1990, Frazier and Jawerth 2, Theorem 5.13 generalized the above duality to homogeneous Triebel-. Q and P always mean the dyadic cubes in Rn, and, for r > 0, we denote by rQ the cube concentric with Q whose each edge is r times as long

Sequence Spaces

Proof of the Main Theorem

Proofs of the Plancherel-Polya Inequalities

Applications