Continuous Characterizations of Besov-Lizorkin-Triebel Spaces and New Interpretations as Coorbits

By Tino Ullrich. 2010

91 citations recorded by the source corpus.

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces (H. Triebel 1983, 1992, and 2006) in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions (tent spaces) and spaces involving the Peetre maximal function to apply the classical coorbit space theory according to Feichtinger and Grochenig (H. G Feichtinger and K. Grochenig 1988, 1989, and 1991). This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a909f1ac6eb52d4afd5cbe0f69aa58ecb20e26ac

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Dachun Yang
2. Wen Yuan
3. Yoshihiro Sawano
4. Tino Ullrich
5. Jun Liu
6. Shohei Nakamura
7. Yiyu Liang
8. Takahiro Noi

In Mathematical Physics

1. Dachun Yang
2. Wen Yuan
3. Yoshihiro Sawano
4. Tino Ullrich
5. Jun Liu
6. Shohei Nakamura
7. Takahiro Noi
8. Yiyu Liang

Citations in the corpus

• New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets (2011, methodology): “In this section, we use the methods from [52] to characterize the homogeneous Besov-type space Ḃ p,q (Rn), the Triebel-Lizorkin-type space Ḟ s,τ p,q (Rn) and their preduals, the BesovHausdorff space BḢ p,q (Rn) and the Triebel-Lizorkin-Hausdorff space FḢ s,τ p,q (Rn) via the local means, the Peetre ”
• Variable Anisotropic Hardy Spaces and Their Applications (2017, methodology): “[43] Y. Liang, Y. Sawano, T. Ullrich, D. Yang and W. Yuan, New characterizations of Besov-Triebel-Lizorkin-Hausdorff spaces including coorbits and wavelets, J. Fourier Anal.”
• Generalized Morrey spaces and trace operator (2016, methodology): “such as Morrey spaces, Herz spaces and Orlicz spaces is to connect these spaces with the Littlewood-Paley decomposition. The idea of replacing Lp(Rn) spaces with other function spaces are expanded in [18, 30, 44, 72]. Such attempts are made for Bσ-function spaces, variable Lebesgue spaces and Orlicz”
• Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications (2018, methodology): “Indeed, by the aforementioned approach from Ullrich [75], via a key lemma (see Lemma 4.14 below) and an auxiliary function gt,∗( f ) (see (4.20) below), we show that the L~p(Rn) quasi-norm of the Lusin area function can be controlled by that of the Littlewood-Paley g-function.”
• A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (2012, unknown): “By the Peetre maximal function characterizations of the Besov space Bs p,q(R) and the Triebel-Lizorkin space F s p,q(R) (see, for example, [93]), we know that, if q ∈ (0,∞], L(Rn) = Lp(Rn) and wj ≡ 2js, then B L,q,a = Bs p,q(R) for all p ∈ (0,∞] and a ∈ (n/p,∞), and F L,q,a = F s p,q(R) for all p ∈ ”
• Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type (2010, background): “(Fv,Y ) The image space RF (Y ) is continuously embedded into L 1/v ∞ (X,µ) and RF belongs to the algebra BY,m, where m is the weight on X ×X associated to v via (3.3).”
• Spaces of Variable Smoothness and Integrability: Characterizations by Local Means and Ball Means of Differences (2011, methodology): “More exhaustive arguments of this type have been used in [54] and [47].”