New applications of Besov-type and Triebel–Lizorkin-type spaces

In this paper, the authors prove that Besov–Morrey spaces are proper subspaces of Besov-type spaces B ˙ p , q s , τ ( R n ) and that Triebel–Lizorkin–Morrey spaces are special cases of Triebel–Lizorkin-type spaces F ˙ p , q s , τ ( R n ) . The authors also establish an equivalent characterization of B ˙ p , q s , τ ( R n ) when τ ∈ [ 0 , 1 / p ) . These Besov-type spaces B ˙ p , q s , τ ( R n ) and Triebel–Lizorkin-type spaces F ˙ p , q s , τ ( R n ) were recently introduced to connect Besov spaces and Triebel–Lizorkin spaces with Q spaces. Moreover, for the spaces B ˙ p , q s , τ ( R n ) and F ˙ p , q s , τ ( R n ) , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos–Torres by taking τ = 0 .