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,tau)(R-n) and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces F-p,q(s,tau)(R-n). The authors also establish an equivalent characterization of B-p,q(s,tau)(R-n) when tau is an element of [0, 1/p). These Besov-type spaces B-p,q(s,tau)(R-n) and Triebel-Lizorkin-type spaces F-p,q(s,tau)(R-n) were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces B-p,q(s,tau)(R-n) and F-p,q(s,tau)(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 tau = 0. (C) 2009 Elsevier Inc. All rights reserved.