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

By Yiyu Liang, Yoshihiro Sawano, Tino Ullrich, Dachun Yang, Wen Yuan. 2011

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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∈(0,∞], s∈ℝ, τ∈[0,∞), and q∈(0,∞], as well as their preduals, the Besov-Hausdorff spaces $B\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-Hausdorff spaces $F\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$, 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 Rauhut (Stud. Math. 180:237–253, 2007) and discretizations via 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∈ℝ, q∈(0,∞] (including BMO(ℝn) when s=0 and q=2), the Q space Qα(ℝn), the Hardy-Hausdorff space HH−α(ℝn) for α∈(0,min{n/2,1}), the Morrey space ${\mathcal{M}}^{u}_{p}({\mathbb{R}}^{n})$ for 1<p≤u<∞, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^{s}_{upq}({\mathbb{R}}^{n})$ for 0<p≤u<∞, s∈ℝ and q∈(0,∞], some of these results are new.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9475a0f65835b233e43eace564dec0e9a84c1ee9

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Citations in the corpus

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood–Paley Characterizations and Real Interpolation (2019, methodology): “[51], which provides one way to control the WX norm of the Lusin area function by the corresponding norm of the Littlewood–Paley g-function, via a key technical lemma (see Lemma 3.”
Variable weak Hardy spaces and their applications (2016, background): “[28], which, via a key and technical lemma (see Lemma 6.”
Generalized Morrey spaces and trace operator (2016, methodology): “Now we can characterize our function spaces by means of the Peetre maximal operator in the spirit of [29, 30, 72].”
Besov-Type Spaces with Variable Smoothness and Integrability (2015, background): “Various properties and equivalent characterizations of Besov-type and Triebel-Lizorkin-type spaces, including smoothness atomic, molecular or wavelet decompositions, characterizations, respectively, via differences, oscillations, Peetre maximal functions, Lusin area functions or g∗ λ functions, have”
Interpolation of Morrey-Campanato and related smoothness spaces (2014, background): “First attempts to a systematic investigation of all these scales are made in [97] and [73, 74]; see also [94, 41].”
A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (2012, unknown): “Then, we have shown in [45] that B p,q,a(R) coincides with B p,q (Rn) as long as a ∈ (np ,∞).”