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 and that Triebel–Lizorkin–Morrey spaces are special cases of Triebel–Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel–Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel–Lizorkin spaces with Q spaces. Moreover, for the spaces and , 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.