Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type

By Holger Rauhut, Tino Ullrich. 2010

85 citations recorded by the source corpus.

Coorbit space theory is an abstract approach to function spaces and their atomic decom- positions. The original theory developed by Feichtinger and Grochenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author (24) that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of var- ious types of interest as coorbits. We obtain several old and new wavelet characterizations based on explicit smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/46ad185a35881763b7bdd786bb28274432078888

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. Shohei Nakamura
5. Takahiro Noi
6. Yiyu Liang
7. M. Rosenthal
8. Tino Ullrich

In Numerical Analysis

1. M. Rosenthal
2. Yoshihiro Sawano
3. Dachun Yang
4. Wen Yuan
5. Yiyu Liang
6. Tino Ullrich
7. Taiji Suzuki

Citations in the corpus

• New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets (2011, methodology): “Notice that we use more general biorthogonal wavelet bases instead of classical orthogonal wavelet bases as it is done for instance in [50, Chapter 3], in case of inhomogeneous classical Besov and Triebel-Lizorkin spaces, and also in [23, 26, 27, 28, 8, 9, 35].”
• Local means, wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces (2013, result): “Wavelet characterizations are also given by Ullrich and Rauhut [61] (for several wavelets) for the Banach spaces, i.e. for 1 ≤ u ≤ p ≤ ∞ and 1 ≤ q ≤ ∞.”
• Generalized Morrey spaces and trace operator (2016, background): “The idea of replacing L(R) spaces with other function spaces are expanded in [18, 30, 44, 72].”
• A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces (2012, unknown): “(i) As is the case with [68], bi-orthogonal systems in Theorem 4.”