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

Abstract

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