Banach algebras of pseudodifferential operators and their almost diagonalization  [ Algèbres de Banach d’opérateurs pseudo-différentiels et leur presque diagonalisation ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 7, p. 2279-2314
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 .
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 .
DOI : https://doi.org/10.5802/aif.2414
Classification:  42C40,  35S05
Mots clés: opérateur pseudodifferentiel, classe de symboles, calcul symbolique, algèbre de Banach, lemme de Wiener
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {7},
     year = {2008},
     pages = {2279-2314},
     doi = {10.5802/aif.2414},
     mrnumber = {2498351},
     zbl = {1168.35050},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_7_2279_0}
}
Gröchenig, Karlheinz; Rzeszotnik, Ziemowit. Banach algebras of pseudodifferential operators and their almost diagonalization. Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2279-2314. doi : 10.5802/aif.2414. http://www.numdam.org/item/AIF_2008__58_7_2279_0/

[1] Auscher, P. Remarks on the local Fourier bases, Wavelets: mathematics and applications (1994), pp. 203-218 (CRC, Boca Raton, FL) | MR 1247517 | Zbl 0882.42026

[2] Balan, R.; Casazza, P. G.; Heil, C.; Landau, Z. Density, overcompleteness, and localization of frames. II. Gabor systems., J. Fourier Anal. Appl., Tome 12 (2006) no. 3, pp. 309-344 | Article | MR 2235170

[3] Baskakov, A. G. Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. i Prilozhen, Tome 24 (1990) no. 3, p. 64-65 | Zbl 0728.47021

[4] Beals, R. Characterization of pseudodifferential operators and applications, Duke Math. J., Tome 44 (1977) no. 1, pp. 45-57 | Article | MR 435933 | Zbl 0353.35088

[5] Bekka, B. Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., Tome 10 (2004) no. 4, pp. 325-349 | Article | MR 2078261 | Zbl 1064.46058

[6] Bochner, S.; Phillips, R. S. Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math., Tome 43 (1942) no. 2, pp. 409-418 | Article | MR 7939 | Zbl 0060.27204

[7] Bonsall, F. F.; Duncan, J. Complete normed algebras, Springer-Verlag, New York (1973) (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80.) | MR 423029 | Zbl 0271.46039

[8] Bony, J.-M.; Chemin, J.-Y. Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, Tome 122 (1994) no. 1, pp. 77-118 | Numdam | MR 1259109 | Zbl 0798.35172

[9] Boulkhemair, A. Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett., Tome 4 (1997) no. 1, pp. 53-67 | MR 1432810 | Zbl 0905.35103

[10] Boulkhemair, A. L 2 estimates for Weyl quantization, J. Funct. Anal., Tome 165 (1999) no. 1, pp. 173-204 | Article | MR 1696697 | Zbl 0934.35217

[11] Brandenburg, L. H. On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl., Tome 50 (1975), pp. 489-510 | Article | MR 377523 | Zbl 0302.46042

[12] Christensen, O. An introduction to frames and Riesz bases, Birkhäuser Boston Inc., Boston, MA, Applied and Numerical Harmonic Analysis (2003) | MR 1946982 | Zbl 1017.42022

[13] Deleeuw, K. An harmonic analysis for operators. I. Formal properties, Illinois J. Math., Tome 19 (1975) no. 4, pp. 593-606 | MR 383002 | Zbl 0313.43018

[14] Feichtinger, H. G. Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, Tome 188 (1979) no. 8-10, pp. 451-471 | MR 599884 | Zbl 0447.43004

[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 751019 | Zbl 0528.43001

[16] Feichtinger, H. G. Generalized amalgams, with applications to Fourier transform, Canad. J. Math., Tome 42 (1990) no. 3, pp. 395-409 | Article | MR 1062738 | Zbl 0733.46014

[17] Feichtinger, H. G. Modulation spaces on locally compact abelian groups, In Proceedings of “International Conference on Wavelets and Applications" 2002, Chennai, India (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., Tome 86 (1989) no. 2, pp. 307-340 | Article | MR 1021139 | Zbl 0691.46011

[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, Wavelets: A tutorial in theory and applications (1992), pp. 359-398 (Academic Press, Boston, MA) | MR 1161258 | Zbl 0849.43003

[20] Feichtinger, H. G.; Gröchenig, K. Gabor frames and time-frequency analysis of distributions, J. Functional Anal., Tome 146 (1997) no. 2, pp. 464-495 | Article | MR 1452000 | Zbl 0887.46017

[21] Folland, G. B. Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ (1989) | MR 983366 | Zbl 0682.43001

[22] Fournier, J. J. F.; Stewart, J. Amalgams of L p and l q , Bull. Amer. Math. Soc. (N.S.), Tome 13 (1985) no. 1, pp. 1-21 | Article | MR 788385 | Zbl 0593.43005

[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 1843717 | Zbl 0966.42020

[25] Gröchenig, K. Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., Tome 10 (2004) no. 2, pp. 105-132 | Article | MR 2054304 | Zbl 1055.42018

[26] Gröchenig, K. Composition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math., Tome 98 (2006), pp. 65 - 82 | Article | MR 2254480 | Zbl pre05256523

[27] Gröchenig, K. Time-frequency analysis of Sjöstrand’s class, Revista Mat. Iberoam, Tome 22 (2006) no. 2, pp. 703-724 (arXiv:math.FA/0409280v1) | Zbl 1127.35089

[28] Gröchenig, K.; Heil, C. Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, Tome 34 (1999) no. 4, pp. 439-457 | Article | MR 1702232 | Zbl 0936.35209

[29] Gröchenig, K.; Heil, C. Modulation spaces as symbol classes for pseudodifferential operators, Wavelets and Their Applications (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., Tome 17 (2004), pp. 1-18 | Article | Zbl 1037.22012

[31] Gröchenig, K.; Samarah, S. Non-linear approximation with local Fourier bases, Constr. Approx., Tome 16 (2000) no. 3, pp. 317-331 | Article | MR 1759892 | Zbl 0973.42025

[32] Hernández, E.; Weiss, G. A first course on wavelets, CRC Press, Boca Raton, FL, Studies in Advanced Mathematics (1996) (With a foreword by Yves Meyer) | MR 1408902 | Zbl 0885.42018

[33] Hörmander, L. The analysis of linear partial differential operators. III, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 274 (1985) (Pseudodifferential operators) | MR 781536 | Zbl 0601.35001

[34] Jaffard, S. Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 7 (1990) no. 5, pp. 461-476 | Numdam | Zbl 0722.15004

[35] Lerner, N.; Morimoto, Y. A Wiener algebra for the Fefferman-Phong inequality, Sémin. Équ. Dériv. Partielles, Seminaire: Equations aux Dérivées Partielles. 2005–2006 (2006) (pages Exp. No. XVII, 12. École Polytech., Palaiseau) | Numdam | MR 2276082 | Zbl 1122.35163

[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 115101 | Zbl 0095.09702

[37] Rochberg, R.; Tachizawa, K. Pseudodifferential operators, Gabor frames, and local trigonometric bases, Gabor analysis and algorithms, Birkhäuser Boston, Boston, MA (1998), pp. 171-192 | MR 1601103 | Zbl 0890.42009

[38] Rudin, W. Functional analysis, McGraw-Hill Book Co., New York, McGraw-Hill Series in Higher Mathematics (1973) | MR 365062 | Zbl 0253.46001

[39] Sjöstrand, J. An algebra of pseudodifferential operators, Math. Res. Lett., Tome 1 (1994) no. 2, pp. 185-192 | MR 1266757 | Zbl 0840.35130

[40] Sjöstrand, J. Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994–1995, pages Exp. No. IV, 21 École Polytech, Palaiseau (1995) | Numdam | MR 1362552 | Zbl 0880.35145

[41] Sjöstrand, J. Pseudodifferential operators and weighted normed symbol spaces, Preprint (2007) (arXiv:0704.1230v1) | MR 2414412

[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 1232192 | Zbl 0821.42001

[43] Stein, E. M.; Weiss, G. Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, Princeton Mathematical Series, No. 32. (1971) | MR 304972 | Zbl 0232.42007

[44] Toft, J. Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble), Tome 51 (2001) no. 5, pp. 1347-1383 | Article | Numdam | MR 1860668 | Zbl 1027.35168

[45] Toft, J. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal., Tome 207 (2004) no. 2, pp. 399-429 | Article | MR 2032995 | Zbl 1083.35148

[46] Toft, J. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom., Tome 26 (2004) no. 1, pp. 73-106 | Article | MR 2054576 | Zbl 1098.47045

[47] Toft, J. Continuity and Schatten properties for pseudo-differential operators on modulation spaces, Oper. Theory Adv. Appl., Birkhäuser, Basel Tome 172 (2007), pp. 173-206 | MR 2308511 | Zbl 1133.35110

[48] Ueberberg, J. Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L p -Theorie, Manuscripta Math., Tome 61 (1988) no. 4, pp. 459-475 | Article | MR 952090 | Zbl 0674.47033