Decompositions of Besov-Hausdorff and Triebel-Lizorkin-Hausdorff Spaces and Their Applications

Abstract

Let $p\in (1,\infty )$, $q\in [1,\infty )$, $s\in \mathbb{R}$ and $\tau \in [0,1-\frac{1}{\max \{p,q\}}]$. In this paper, the authors establish the $\phi$-transform characterizations of Besov-Hausdorff spaces $B{\dot{H}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ and Triebel-Lizorkin-Hausdorff spaces $F{\dot{H}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ ($q>1$); as applications, the authors then establish their embedding properties (which on $B{\dot{H}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ is also sharp), smooth atomic and molecular decomposition characterizations for suitable $\tau$. 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 $B{\dot{H}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ and $F{\dot{H}}_{p,q}^{s,\tau }({\mathbb{R}}^n)$ ($q>1$), which generalize the corresponding classical results on homogeneous Besov and Triebel-Lizorkin spaces when $p\in (1,\infty )$ and $q\in [1,\infty )$ by taking $\tau =0$.

arXiv comment30 pages, J. Math. Anal. Appl. (to appear).