Decompositions of Besov–Hausdorff and Triebel–Lizorkin–Hausdorff spaces and their applications

Let p∈(1,∞), q∈[1,∞), s∈R and . In this paper, the authors establish the φ-transform characterizations of Besov–Hausdorff spaces and Triebel–Lizorkin–Hausdorff spaces (q>1); as applications, the authors then establish their embedding properties (which on is also sharp), smooth atomic and molecular decomposition characterizations for suitable τ. Moreover, using their atomic and molecular decomposition characterizations, the authors investigate the trace properties and the boundedness of pseudo-differential operators with homogeneous symbols in and (q>1), which generalize the corresponding classical results on homogeneous Besov and Triebel–Lizorkin spaces when p∈(1,∞) and q∈[1,∞) by taking τ=0.