Banach algebras of pseudodifferential operators and their almost diagonalization

Gröchenig, Karlheinz; Rzeszotnik, Ziemowit

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 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 $\Lambda$ correspond une classe de symboles $M^{\infty ,𝒜}$. Chaque opérateur pseudo-différentiel dans $M^{\infty ,𝒜}$ 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 $M^{\infty ,𝒜}$ sont bornés sur $L^2\left({ℝ}^d\right)$ 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 $\Lambda$ we associate a symbol class $M^{\infty ,𝒜}$. Then every operator with a symbol in $M^{\infty ,𝒜}$ 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 $L^2\left({ℝ}^d\right)$. 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 $S_{0,0}^0$.

Classification : 42C40, 35S05
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

[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, 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., 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, Wavelets: A tutorial in theory and applications (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, 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., 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, Seminaire: Equations aux Dérivées Partielles. 2005–2006 (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, Gabor analysis and algorithms, Birkhäuser Boston, Boston, MA (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, pages Exp. No. IV, 21 École Polytech, Palaiseau (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, Oper. Theory Adv. Appl., Birkhäuser, Basel (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