Research PaperNo Access

Discrete decomposition of mixed-norm $α$ -modulation spaces

Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, DK-9220 Aalborg East, Denmark

https://doi.org/10.1142/S0219691324500383Cited by:0 (Source: Crossref)

Abstract

We study a class of almost diagonal matrices compatible with the mixed-norm $α$ -modulation spaces $M_{p, q}^{s, α} (ℝ^{n})$ , $α \in [0, 1]$ , introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm $α$ -modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm $α$ -modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm $α$ -modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm $α$ -modulation spaces.

Keywords:

AMSC: 47G30, 46E35, 47B38

References

1. N. Antonić and I. Ivec , On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces, J. Math. Anal. Appl. 433(1) (2016) 176–199. Crossref, Web of Science, Google Scholar
2. R. J. Bagby , An extended inequality for the maximal function, Proc. Amer. Math. Soc. 48 (1975) 419–422. Crossref, Web of Science, Google Scholar
3. A. Benedek and R. Panzone , The space $L^{p}$ , with mixed norm, Duke Math. J. 28 (1961) 301–324. Crossref, Web of Science, Google Scholar
4. A. Bényi and M. Bownik , Anisotropic classes of homogeneous pseudodifferential symbols, Studia Math. 200(1) (2010) 41–66. Crossref, Web of Science, Google Scholar
5. L. Borup , Pseudodifferential operators on $α$ -modulation spaces, J. Funct. Spaces Appl. 2(2) (2004) 107–123. Crossref, Google Scholar
6. L. Borup and M. Nielsen , Banach frames for multivariate $α$ -modulation spaces, J. Math. Anal. Appl. 321(2) (2006) 880–895. Crossref, Google Scholar
7. L. Borup and M. Nielsen , Frame decomposition of decomposition spaces, J. Fourier Anal. Appl. 13(1) (2007) 39–70. Crossref, Web of Science, Google Scholar
8. L. Borup and M. Nielsen , On anisotropic Triebel–Lizorkin type spaces, with applications to the study of pseudo-differential operators, J. Funct. Spaces Appl. 6(2) (2008) 107–154. Crossref, Google Scholar
9. M. Bownik , Atomic and molecular decompositions of anisotropic Besov spaces, Math. Z. 250(3) (2005) 539–571. Crossref, Web of Science, Google Scholar
10. M. Bownik and K.-P. Ho , Atomic and molecular decompositions of anisotropic Triebel–Lizorkin spaces, Trans. Amer. Math. Soc. 358(4) (2006) 1469–1510. Crossref, Web of Science, Google Scholar
11. G. Cleanthous and A. G. Georgiadis , Mixed-norm $α$ -modulation spaces, Trans. Amer. Math. Soc. 373(5) (2020) 3323–3356. Crossref, Google Scholar
12. G. Cleanthous and A. G. Georgiadis . Product ( $α$ 1, $α$ 2)-modulation spaces, Sci. China Math. 65(8) (2022) 1599–1640. Crossref, Google Scholar
13. G. Cleanthous, A. G. Georgiadis and M. Nielsen , Discrete decomposition of homogeneous mixed-norm Besov spaces, in Functional Analysis, Harmonic Analysis, and Image Processing: A Collection of Papers in Honor of Björn Jawerth, Contemporary Mathematics, Vol. 693 (American Mathematical Society, Providence, 2017), pp. 167–184. Crossref, Google Scholar
14. G. Cleanthous, A. G. Georgiadis and M. Nielsen , Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal. 47(2) (2019) 447–480. Crossref, Web of Science, Google Scholar
15. H. G. Feichtinger , Banach spaces of distributions defined by decomposition methods. II, Math. Nachr. 132 (1987) 207–237. Crossref, Google Scholar
16. H. G. Feichtinger , Modulation spaces of locally compact abelian groups, in Proc. Int. Conf. Wavelets and Applications, eds. R. Radha, M. Krishna and S. Thangavelu (Allied Publishers, New Delhi, 2003), pp. 1–56. Google Scholar
17. H. G. Feichtinger and P. Gröbner , Banach spaces of distributions defined by decomposition methods, I, Math. Nachr. 123 (1985) 97–120. Crossref, Web of Science, Google Scholar
18. M. Frazier and B. Jawerth , Decomposition of Besov spaces, Indiana Univ. Math. J. 34(4) (1985) 777–799. Crossref, Web of Science, Google Scholar
19. M. Frazier and B. Jawerth , A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93(1) (1990) 34–170. Crossref, Web of Science, Google Scholar
20. A. G. Georgiadis and M. Nielsen , Pseudodifferential operators on mixed-norm Besov and Triebel–Lizorkin spaces, Math. Nachr. 289(16) (2016) 2019–2036. Crossref, Web of Science, Google Scholar
21. A. G. Georgiadis and M. Nielsen , Pseudodifferential operators on spaces of distributions associated with non-negative self-adjoint operators, J. Fourier Anal. Appl. 23(2) (2017) 344–378. Crossref, Web of Science, Google Scholar
22. P. Gröbner, Banachräume glatter Funktionen und Zerlegungsmethoden, PhD thesis, University of Vienna (1992). Google Scholar
23. J. Johnsen and W. Sickel , A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin–Triebel spaces with mixed norms, J. Funct. Spaces Appl. 5(2) (2007) 183–198. Crossref, Google Scholar
24. J. Johnsen and W. Sickel , On the trace problem for Lizorkin–Triebel spaces with mixed norms, Math. Nachr. 281(5) (2008) 669–696. Crossref, Web of Science, Google Scholar
25. G. Kyriazis and P. Petrushev , On the construction of frames for Triebel–Lizorkin and Besov spaces, Proc. Amer. Math. Soc. 134(6) (2006) 1759–1770. Crossref, Web of Science, Google Scholar
26. B. Li, M. Bownik and D. Yang , Littlewood–Paley characterization and duality of weighted anisotropic product Hardy spaces, J. Funct. Anal. 266(5) (2014) 2611–2661. Crossref, Web of Science, Google Scholar
27. M. Nielsen , Pseudodifferential operators on mixed-norm $α$ -modulation spaces, Rocky Mountain J. Math. 53(3) (2023) 875–887. Crossref, Google Scholar
28. M. Nielsen and K. N. Rasmussen , Compactly supported frames for decomposition spaces, J. Fourier Anal. Appl. 18(1) (2012) 87–117. Crossref, Web of Science, Google Scholar