Generalized Carleson Measure Spaces and Their Applications

Abstract

We introduce the generalized Carleson measure spaces CM${\text{O}}_r^{\alpha ,q}$ that extend BMO. Using Frazier and Jawerth's $\phi$-transform and sequence spaces, we show that, for $\alpha \in \mathbb{R}$ and , the duals of homogeneous Triebel-Lizorkin spaces ${\dot{F}}_p^{\alpha ,q}$ for and are CM${\text{O}}_{(q\text{'}/p)-(q\text{'}/q)}^{-\alpha ,q\text{'}}$ and CM${\text{O}}_r^{-\alpha +(n/p)-n,\infty }$ (for any $r\in \mathbb{R}$), 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.