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

AbstractLet p∈(1,∞), q∈[1,∞), s∈R and τ∈[0,1−1max{p,q}]. In this paper, the authors establish the φ-transform characterizations of Besov–Hausdorff spaces BH˙p,qs,τ(Rn) and Triebel–Lizorkin–Hausdorff spaces FH˙p,qs,τ(Rn) (q>1); as applications, the authors then establish their embedding properties (which on BH˙p,qs,τ(Rn) 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 BH˙p,qs,τ(Rn) and FH˙p,qs,τ(Rn) (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