Generalized Carleson Measure Spaces and Their Applications

We introduce the generalized Carleson measure spaces CM O r α , q that extend BMO. Using Frazier and Jawerth's φ -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 CM O ( q ' / p ) - ( q ' / q ) - α , q ' and CM O 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.