New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb R}^n)$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb R}^n)$ with $p\in (0,\infty]$, $s\in{\mathbb R}$, $\tau\in [0,\infty)$, and $q\in (0,\infty]$, as well as their preduals, the Besov-Hausdorff spaces $B\dot{H}^{s,\tau}_{p,q}(\R^n)$ and Triebel-Lizorkin-Hausdorff spaces $F\dot{H}^{s,\tau}_{p,q}(\R^n)$, in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of H. Rauhut in [Studia Math. 180 (2007), 237-253] and discretizations via the biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot F^s_{\infty,q}({\mathbb R}^n)$ for $s\in{\mathbb R}$, $q\in (0,\infty]$ (including $\mathop\mathrm{BMO} ({\mathbb R}^n)$ when $s=0$, $q=2$), the $Q$ space $Q_\alpha ({\mathbb R}^n)$, the Hardy-Hausdorff space $HH_{-\alpha}({\mathbb R}^n)$ for $\alpha\in (0,\min\{\frac n2,1\})$, the Morrey space ${\mathcal M}^u_p({\mathbb R}^n)$ for $1<p\le u<\infty$, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^s_{upq}({\mathbb R}^n)$ for $0<p\le u<\infty$, $s\in{\mathbb R}$ and $q\in(0,\infty]$, some of these results are new.