Banach algebras of pseudodifferential operators and their almost diagonalization
Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2279-2314.

We define new symbol classes for pseudodifferential operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra 𝒜 over a lattice Λ we associate a symbol class M ,𝒜 . Then every operator with a symbol in M ,𝒜 is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra 𝒜. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on L 2 ( d ). If a version of Wiener’s lemma holds for 𝒜, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sjöstrand’s class and yields a new proof of a theorem of Beals about the Hörmander class S 0,0 0 .

Nous étudions une nouvelle classe de symboles pour les opérateurs pseudo-différentiels et leurs calculs symboliques. À chaque algèbre 𝒜 commutative par rapport aux convolutions sur un réseau Λ correspond une classe de symboles M ,𝒜 . Chaque opérateur pseudo-différentiel dans M ,𝒜 est presque diagonale par rapport aux états cohérents, et sa décroissance hors de la diagonale est décrite par l’algèbre 𝒜. Les opérateurs pseudo-différentiels avec des symboles dans M ,𝒜 sont bornés sur L 2 ( d ) et constituent une algèbre de Banach. Si une version du lemme de Wiener s’applique à 𝒜, alors l’algèbre d’opérateurs pseudo-différentiels est fermée par rapport à l’inversion des opérateurs. La théorie contient comme un cas spécial la théorie de J. Sjöstrand et fournit une nouvelle démonstration d’un théorème de Beals sur les symboles de Hörmander dans S 0,0 0 .

DOI: 10.5802/aif.2414
Classification: 42C40, 35S05
Keywords: Pseudodifferential operators, symbol class, symbolic calculus, Banach algebra, inverse-closedness, Wiener’s Lemma
Gröchenig, Karlheinz 1; Rzeszotnik, Ziemowit 2

1 University of Vienna Faculty of Mathematics Nordbergstrasse 15 1090 Wien (Austria)
2 University of Wroclaw Mathematical Institute Pl. Grunwaldzki 2/4 50-384 Wroclaw (Poland)
@article{AIF_2008__58_7_2279_0,
     author = {Gr\"ochenig, Karlheinz and Rzeszotnik, Ziemowit},
     title = {Banach algebras of pseudodifferential operators and their almost diagonalization},
     journal = {Annales de l'Institut Fourier},
     pages = {2279--2314},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {7},
     year = {2008},
     doi = {10.5802/aif.2414},
     mrnumber = {2498351},
     zbl = {1168.35050},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2414/}
}
TY  - JOUR
AU  - Gröchenig, Karlheinz
AU  - Rzeszotnik, Ziemowit
TI  - Banach algebras of pseudodifferential operators and their almost diagonalization
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 2279
EP  - 2314
VL  - 58
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2414/
DO  - 10.5802/aif.2414
LA  - en
ID  - AIF_2008__58_7_2279_0
ER  - 
%0 Journal Article
%A Gröchenig, Karlheinz
%A Rzeszotnik, Ziemowit
%T Banach algebras of pseudodifferential operators and their almost diagonalization
%J Annales de l'Institut Fourier
%D 2008
%P 2279-2314
%V 58
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2414/
%R 10.5802/aif.2414
%G en
%F AIF_2008__58_7_2279_0
Gröchenig, Karlheinz; Rzeszotnik, Ziemowit. Banach algebras of pseudodifferential operators and their almost diagonalization. Annales de l'Institut Fourier, Volume 58 (2008) no. 7, pp. 2279-2314. doi : 10.5802/aif.2414. https://aif.centre-mersenne.org/articles/10.5802/aif.2414/

[1] Auscher, P. Remarks on the local Fourier bases, Wavelets: mathematics and applications (1994), pp. 203-218 (CRC, Boca Raton, FL) | MR | Zbl

[2] Balan, R.; Casazza, P. G.; Heil, C.; Landau, Z. Density, overcompleteness, and localization of frames. II. Gabor systems., J. Fourier Anal. Appl., Volume 12 (2006) no. 3, pp. 309-344 | DOI | MR

[3] Baskakov, A. G. Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. i Prilozhen, Volume 24 (1990) no. 3, pp. 64-65 | Zbl

[4] Beals, R. Characterization of pseudodifferential operators and applications, Duke Math. J., Volume 44 (1977) no. 1, pp. 45-57 | DOI | MR | Zbl

[5] Bekka, B. Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., Volume 10 (2004) no. 4, pp. 325-349 | DOI | MR | Zbl

[6] Bochner, S.; Phillips, R. S. Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math., Volume 43 (1942) no. 2, pp. 409-418 | DOI | MR | Zbl

[7] Bonsall, F. F.; Duncan, J. Complete normed algebras, Springer-Verlag, New York, 1973 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80.) | MR | Zbl

[8] Bony, J.-M.; Chemin, J.-Y. Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, Volume 122 (1994) no. 1, pp. 77-118 | Numdam | MR | Zbl

[9] Boulkhemair, A. Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., Volume 4 (1997) no. 1, pp. 53-67 | MR | Zbl

[10] Boulkhemair, A. L 2 estimates for Weyl quantization, J. Funct. Anal., Volume 165 (1999) no. 1, pp. 173-204 | DOI | MR | Zbl

[11] Brandenburg, L. H. On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl., Volume 50 (1975), pp. 489-510 | DOI | MR | Zbl

[12] Christensen, O. An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston Inc., Boston, MA, 2003 | MR | Zbl

[13] deLeeuw, K. An harmonic analysis for operators. I. Formal properties, Illinois J. Math., Volume 19 (1975) no. 4, pp. 593-606 | MR | Zbl

[14] Feichtinger, H. G. Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, Volume 188 (1979) no. 8-10, pp. 451-471 | MR | Zbl

[15] Feichtinger, H. G. Banach convolution algebras of Wiener type, In Functions, series, operators, Vol. I, II (Budapest, 1980) (1983), pp. 509-524 (North-Holland, Amsterdam) | MR | Zbl

[16] Feichtinger, H. G. Generalized amalgams, with applications to Fourier transform, Canad. J. Math., Volume 42 (1990) no. 3, pp. 395-409 | DOI | MR | Zbl

[17] Feichtinger, H. G. Modulation spaces on locally compact abelian groups, In Proceedings of “International Conference on Wavelets and Applications" 2002 (2003), pp. 99-140 (Updated version of a technical report, University of Vienna, 1983)

[18] Feichtinger, H. G.; Gröchenig, K. Banach spaces related to integrable group representations and their atomic decompositions. I, J. Functional Anal., Volume 86 (1989) no. 2, pp. 307-340 | DOI | MR | Zbl

[19] Feichtinger, H. G.; Gröchenig, K. Gabor wavelets and the Heisenberg group: Gabor expansions and short time fourier transform from the group theoretical point of view (1992), pp. 359-398 (Academic Press, Boston, MA) | MR | Zbl

[20] Feichtinger, H. G.; Gröchenig, K. Gabor frames and time-frequency analysis of distributions, J. Functional Anal., Volume 146 (1997) no. 2, pp. 464-495 | DOI | MR | Zbl

[21] Folland, G. B. Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl

[22] Fournier, J. J. F.; Stewart, J. Amalgams of L p and l q , Bull. Amer. Math. Soc. (N.S.), Volume 13 (1985) no. 1, pp. 1-21 | DOI | MR | Zbl

[23] Gel’fand, I.; Raikov, D.; Shilov, G. Commutative normed rings, Chelsea Publishing Co., New York, 1964

[24] Gröchenig, K. Foundations of time-frequency analysis, Birkhäuser Boston Inc., Boston, MA, 2001 | MR | Zbl

[25] Gröchenig, K. Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., Volume 10 (2004) no. 2, pp. 105-132 | DOI | MR | Zbl

[26] Gröchenig, K. Composition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math., Volume 98 (2006), pp. 65 - 82 | DOI | MR

[27] Gröchenig, K. Time-frequency analysis of Sjöstrand’s class, Revista Mat. Iberoam, Volume 22 (2006) no. 2, pp. 703-724 (arXiv:math.FA/0409280v1) | Zbl

[28] Gröchenig, K.; Heil, C. Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, Volume 34 (1999) no. 4, pp. 439-457 | DOI | MR | Zbl

[29] Gröchenig, K.; Heil, C. Modulation spaces as symbol classes for pseudodifferential operators (2003), pp. 151-170 (Allied Publishers, Chennai)

[30] Gröchenig, K.; Leinert, M. Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., Volume 17 (2004), pp. 1-18 | DOI | Zbl

[31] Gröchenig, K.; Samarah, S. Non-linear approximation with local Fourier bases, Constr. Approx., Volume 16 (2000) no. 3, pp. 317-331 | DOI | MR | Zbl

[32] Hernández, E.; Weiss, G. A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996 (With a foreword by Yves Meyer) | MR | Zbl

[33] Hörmander, L. The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1985 (Pseudodifferential operators) | MR | Zbl

[34] Jaffard, S. Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 7 (1990) no. 5, pp. 461-476 | Numdam | Zbl

[35] Lerner, N.; Morimoto, Y. A Wiener algebra for the Fefferman-Phong inequality, Sémin. Équ. Dériv. Partielles (2006) (pages Exp. No. XVII, 12. École Polytech., Palaiseau) | Numdam | MR | Zbl

[36] Rickart, C. E. General theory of Banach algebras, The University Series in Higher Mathematics. D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960 | MR | Zbl

[37] Rochberg, R.; Tachizawa, K. Pseudodifferential operators, Gabor frames, and local trigonometric bases (1998), pp. 171-192 | MR | Zbl

[38] Rudin, W. Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973 | MR | Zbl

[39] Sjöstrand, J. An algebra of pseudodifferential operators, Math. Res. Lett., Volume 1 (1994) no. 2, pp. 185-192 | MR | Zbl

[40] Sjöstrand, J. Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994–1995 (1995) | Numdam | MR | Zbl

[41] Sjöstrand, J. Pseudodifferential operators and weighted normed symbol spaces, Preprint, 2007 (arXiv:0704.1230v1) | MR

[42] Stein, E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993 (With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III) | MR | Zbl

[43] Stein, E. M.; Weiss, G. Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32., Princeton Univ. Press, Princeton, NJ, 1971 | MR | Zbl

[44] Toft, J. Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble), Volume 51 (2001) no. 5, pp. 1347-1383 | DOI | Numdam | MR | Zbl

[45] Toft, J. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal., Volume 207 (2004) no. 2, pp. 399-429 | DOI | MR | Zbl

[46] Toft, J. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom., Volume 26 (2004) no. 1, pp. 73-106 | DOI | MR | Zbl

[47] Toft, J. Continuity and Schatten properties for pseudo-differential operators on modulation spaces, 172 (2007), pp. 173-206 | MR | Zbl

[48] Ueberberg, J. Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L p -Theorie, Manuscripta Math., Volume 61 (1988) no. 4, pp. 459-475 | DOI | MR | Zbl

Cited by Sources: