Generalized carleson measure spaces and their applications

We introduce the generalized Carleson measure spaces CMO _r ^α,q that extend BMO. Using Frazier and Jawerth's ψ-transform and sequence spaces, we show that, for α ∈ ℝ and 0 < p ≤ 1, the duals of homogeneous Triebel-Lizorkin spaces F _p ^α,q for 1 < q < ∞ and 0 < q < 1 are CMO ^-αq' _{(q'/p)-(q'/q)} and CMO _r ^{-α+ (n/p)-n,∞} (for any r ∈ 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.