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 over a lattice we associate a symbol class . Then every operator with a symbol in 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 . Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on . If a version of Wiener’s lemma holds for , 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 .
Nous étudions une nouvelle classe de symboles pour les opérateurs pseudo-différentiels et leurs calculs symboliques. À chaque algèbre commutative par rapport aux convolutions sur un réseau correspond une classe de symboles . Chaque opérateur pseudo-différentiel dans est presque diagonale par rapport aux états cohérents, et sa décroissance hors de la diagonale est décrite par l’algèbre . Les opérateurs pseudo-différentiels avec des symboles dans sont bornés sur et constituent une algèbre de Banach. Si une version du lemme de Wiener s’applique à , 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 .
Keywords: Pseudodifferential operators, symbol class, symbolic calculus, Banach algebra, inverse-closedness, Wiener’s Lemma
Mot 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}, pages = {2279--2314}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {58}, number = {7}, year = {2008}, doi = {10.5802/aif.2414}, zbl = {1168.35050}, mrnumber = {2498351}, language = {en}, url = {http://archive.numdam.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 - http://archive.numdam.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 http://archive.numdam.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. http://archive.numdam.org/articles/10.5802/aif.2414/
[1] Remarks on the local Fourier bases, Wavelets: mathematics and applications (1994), pp. 203-218 (CRC, Boca Raton, FL) | MR | Zbl
[2] Density, overcompleteness, and localization of frames. II. Gabor systems., J. Fourier Anal. Appl., Volume 12 (2006) no. 3, pp. 309-344 | DOI | MR
[3] 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] Characterization of pseudodifferential operators and applications, Duke Math. J., Volume 44 (1977) no. 1, pp. 45-57 | DOI | MR | Zbl
[5] 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] Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math., Volume 43 (1942) no. 2, pp. 409-418 | DOI | MR | Zbl
[7] Complete normed algebras, Springer-Verlag, New York, 1973 (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80.) | MR | Zbl
[8] Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France, Volume 122 (1994) no. 1, pp. 77-118 | Numdam | MR | Zbl
[9] 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] estimates for Weyl quantization, J. Funct. Anal., Volume 165 (1999) no. 1, pp. 173-204 | DOI | MR | Zbl
[11] On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl., Volume 50 (1975), pp. 489-510 | DOI | MR | Zbl
[12] An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston Inc., Boston, MA, 2003 | MR | Zbl
[13] An harmonic analysis for operators. I. Formal properties, Illinois J. Math., Volume 19 (1975) no. 4, pp. 593-606 | MR | Zbl
[14] Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, Volume 188 (1979) no. 8-10, pp. 451-471 | MR | Zbl
[15] 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] Generalized amalgams, with applications to Fourier transform, Canad. J. Math., Volume 42 (1990) no. 3, pp. 395-409 | DOI | MR | Zbl
[17] 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] 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] 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 | Zbl
[20] Gabor frames and time-frequency analysis of distributions, J. Functional Anal., Volume 146 (1997) no. 2, pp. 464-495 | DOI | MR | Zbl
[21] Harmonic Analysis in Phase Space, Princeton Univ. Press, Princeton, NJ, 1989 | MR | Zbl
[22] Amalgams of and , Bull. Amer. Math. Soc. (N.S.), Volume 13 (1985) no. 1, pp. 1-21 | DOI | MR | Zbl
[23] Commutative normed rings, Chelsea Publishing Co., New York, 1964
[24] Foundations of time-frequency analysis, Birkhäuser Boston Inc., Boston, MA, 2001 | MR | Zbl
[25] 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] Composition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math., Volume 98 (2006), pp. 65 - 82 | DOI | MR
[27] Time-frequency analysis of Sjöstrand’s class, Revista Mat. Iberoam, Volume 22 (2006) no. 2, pp. 703-724 (arXiv:math.FA/0409280v1) | Zbl
[28] Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory, Volume 34 (1999) no. 4, pp. 439-457 | DOI | MR | Zbl
[29] Modulation spaces as symbol classes for pseudodifferential operators, Wavelets and Their Applications (2003), pp. 151-170 (Allied Publishers, Chennai)
[30] Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc., Volume 17 (2004), pp. 1-18 | DOI | Zbl
[31] Non-linear approximation with local Fourier bases, Constr. Approx., Volume 16 (2000) no. 3, pp. 317-331 | DOI | MR | Zbl
[32] A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996 (With a foreword by Yves Meyer) | MR | Zbl
[33] 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] 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] 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 | Zbl
[36] 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] Pseudodifferential operators, Gabor frames, and local trigonometric bases, Gabor analysis and algorithms, Birkhäuser Boston, Boston, MA (1998), pp. 171-192 | MR | Zbl
[38] Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York, 1973 | MR | Zbl
[39] An algebra of pseudodifferential operators, Math. Res. Lett., Volume 1 (1994) no. 2, pp. 185-192 | MR | Zbl
[40] 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 | Zbl
[41] Pseudodifferential operators and weighted normed symbol spaces, Preprint, 2007 (arXiv:0704.1230v1) | MR
[42] 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] Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32., Princeton Univ. Press, Princeton, NJ, 1971 | MR | Zbl
[44] 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] 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] 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] Continuity and Schatten properties for pseudo-differential operators on modulation spaces, 172, Oper. Theory Adv. Appl., Birkhäuser, Basel (2007), pp. 173-206 | MR | Zbl
[48] Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der -Theorie, Manuscripta Math., Volume 61 (1988) no. 4, pp. 459-475 | DOI | MR | Zbl
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