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

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 $\varphi$-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 $\tau=0$.Comment: 30 pages, J. Math. Anal. Appl. (to appear)

Bounded compositions on scaling invariant Besov spaces

For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $\varphi : \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ \varphi$ is bounded on the homogeneous, scaling invariant Besov space $\dot{B}^s_{n/s,q}(\real^n)$, where the emphasis is on the case $q\not=n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces $\dot{F}^s_{n/s,q}(\real^n)$ with $0 < s < 1$ and $0 < q \leq \infty$.Comment: 20 pages; corrected typos, simplified assumptions for Proposition 3.5 and Theorem 3.