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

Abstract

In this paper, the authors establish new characterizations of the recently introduced Besov-type spacesḂ s,τ p,q (R n ) and Triebel-Lizorkin-type spacesḞ s,τ p,q (R n ) with p ∈ (0, ∞], s ∈ R, τ ∈ [0, ∞), and q ∈ (0, ∞], as well as their preduals, the Besov-Hausdorff spaces BḢ s,τ p,q (R n ) and Triebel-Lizorkin-Hausdorff spaces FḢ s,τ 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][238][239][240][241][242][243][244][245][246][247][248][249][250][251][252][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 asḞ s ∞,q (R n ) for s ∈ R, q ∈ (0, ∞] (including BMO(R n ) when s = 0 and q = 2), the Q space Q α (R n ), the Hardy-Hausdorff space HH −α (R n ) for α ∈ (0, min{n/2, 1}), the Morrey space M u p (R n ) for 1 < p ≤ u < ∞, and the Triebel-Lizorkin-Morrey spaceĖ s upq (R n ) for 0 < p ≤ u < ∞, s ∈ R and q ∈ (0, ∞], some of these results are new. investigated in [57,58,43,61,62]. These spaces establish the connection between the classical Besov-Triebel-Lizorkin spaces (see, for example, [47,22,49]), the Q spaces in [17], and the Hardy-Hausdorff spaces in [15], which has been posed as an open question in [15]. It was shown in [57,58,43] that these spaces unify and generalize many classical function spaces including classical Besov and Triebel-Lizorkin spaces, Triebel-Lizorkin-Morrey spaces (see, for example, [51,42,41,43]), Q spaces, and Hardy-Hausdorff spaces (see also [54,55]).These spaces are usually defined through building blocks constructed out of a dyadic decomposition of unity on the Fourier side. Several of the mentioned applications make it necessary to use more general convolution kernels for the definition of the spaces, especially the so-called local means of a function are of particular interest. By the pioneering work of Bui, Paluszyński and Taibleson [10,11], and later by Rychkov [37,38,39], we know that one rather uses kernels satisfying the Tauberian conditions (see (3.2) below), for characterizing classical Besov-Triebel-Lizorkin spaces. This also applies to the setting of the spaceṡ B s,τ p,q (R n ),Ḟ s,τ p,q (R n ), BḢ s,τ p,q (R n ) and FḢ s,τ p,q (R n ) considered here, which represents one of the main results of the paper; see Theorems 3.2, 3.6, 3.12 and 3.15 below. To establish these results we use a key estimate (see Lemma 3.5 below) which was obtained in [52] by a variant of a method from Rychkov [38,39] and originally from Strömberg and Torchinsky [46, Chapter 5]. In addition we make use of a certain involved decomposition of R n into proper sub-cubes in combination with some localized modifications of the approaches used for Besov and Triebel-Lizorkin spaces. For the technically more difficult Hausdorff type spaces BḢ s,τ p,q ...

Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

“…When p i = q i , i = 0, 1, the Triebel-Lizorkin-Morrey Lorentz spaces reduce to the homogeneous Triebel-Lizorkin-Morrey spaces. The family of inhomogeneous Triebel-Lizorkin-Morrey spaces was studied in [25], [42], [43], [45], [49], [51], [53][54][55]. We have the following result for the embedding of Triebel-Lizorkin-Morrey spaces.…”

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions, local means and non‐smooth atomic decompositions

“…Remark The case where is constant and for all cubes Q and is covered by [, Lemma 2.9]. If is also constant we refer to [, Lemma 2.3] and [, Lemma 2.1].…”

The radial lemma of Strauss in the context of Morrey spaces

“…The extension to quasi-Banach spaces has been done in [47,48]; see also [49,22]. For a first systematic study of these spaces, we refer to the lecture note [53]; see also [31,32,51].…”