Banach algebras of pseudodifferential operators and their almost diagonalization

Karlheinz Gröchenig[1]; Ziemowit Rzeszotnik[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)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 7, page 2279-2314
  • ISSN: 0373-0956

Abstract

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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 ( ℝ 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 .

How to cite

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Gröchenig, Karlheinz, and Rzeszotnik, Ziemowit. "Banach algebras of pseudodifferential operators and their almost diagonalization." Annales de l’institut Fourier 58.7 (2008): 2279-2314. <http://eudml.org/doc/10378>.

@article{Gröchenig2008,
abstract = {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 $\mathcal\{A\} $ over a lattice $\Lambda $ we associate a symbol class $M^\{\infty , \mathcal\{A\} \} $. Then every operator with a symbol in $M^\{\infty ,\mathcal\{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 $\mathcal\{A\} $. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2(\{\mathbb\{R\}\}^d) $. If a version of Wiener’s lemma holds for $\mathcal\{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 $S^0_\{0,0\}$.},
affiliation = {University of Vienna Faculty of Mathematics Nordbergstrasse 15 1090 Wien (Austria); University of Wroclaw Mathematical Institute Pl. Grunwaldzki 2/4 50-384 Wroclaw (Poland)},
author = {Gröchenig, Karlheinz, Rzeszotnik, Ziemowit},
journal = {Annales de l’institut Fourier},
keywords = {Pseudodifferential operators; symbol class; symbolic calculus; Banach algebra; inverse-closedness; Wiener’s Lemma; pseudodifferential operators; Wiener's Lemma},
language = {eng},
number = {7},
pages = {2279-2314},
publisher = {Association des Annales de l’institut Fourier},
title = {Banach algebras of pseudodifferential operators and their almost diagonalization},
url = {http://eudml.org/doc/10378},
volume = {58},
year = {2008},
}

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
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 7
SP - 2279
EP - 2314
AB - 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 $\mathcal{A} $ over a lattice $\Lambda $ we associate a symbol class $M^{\infty , \mathcal{A} } $. Then every operator with a symbol in $M^{\infty ,\mathcal{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 $\mathcal{A} $. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2({\mathbb{R}}^d) $. If a version of Wiener’s lemma holds for $\mathcal{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 $S^0_{0,0}$.
LA - eng
KW - Pseudodifferential operators; symbol class; symbolic calculus; Banach algebra; inverse-closedness; Wiener’s Lemma; pseudodifferential operators; Wiener's Lemma
UR - http://eudml.org/doc/10378
ER -

References

top
  1. P. Auscher, Remarks on the local Fourier bases, Wavelets: mathematics and applications (1994), 203-218 Zbl0882.42026MR1247517
  2. R. Balan, P. G. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames. II. Gabor systems., J. Fourier Anal. Appl. 12 (2006), 309-344 Zbl1097.42022MR2235170
  3. A. G. Baskakov, Wiener’s theorem and asymptotic estimates for elements of inverse matrices, Funktsional. Anal. i Prilozhen 24 (1990), 64-65 Zbl0728.47021
  4. R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45-57 Zbl0353.35088MR435933
  5. B. Bekka, Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl. 10 (2004), 325-349 Zbl1064.46058MR2078261
  6. S. Bochner, R. S. Phillips, Absolutely convergent Fourier expansions for non-commutative normed rings, Ann. of Math. 43 (1942), 409-418 Zbl0060.27204MR7939
  7. F. F. Bonsall, J. Duncan, Complete normed algebras, (1973), Springer-Verlag, New York Zbl0271.46039MR423029
  8. J.-M. Bony, J.-Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. Soc. Math. France 122 (1994), 77-118 Zbl0798.35172MR1259109
  9. A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. Lett. 4 (1997), 53-67 Zbl0905.35103MR1432810
  10. A. Boulkhemair, L 2 estimates for Weyl quantization, J. Funct. Anal. 165 (1999), 173-204 Zbl0934.35217MR1696697
  11. L. H. Brandenburg, On identifying the maximal ideals in Banach algebras, J. Math. Anal. Appl. 50 (1975), 489-510 Zbl0302.46042MR377523
  12. O. Christensen, An introduction to frames and Riesz bases, (2003), Birkhäuser Boston Inc., Boston, MA Zbl1017.42022MR1946982
  13. K. deLeeuw, An harmonic analysis for operators. I. Formal properties, Illinois J. Math. 19 (1975), 593-606 Zbl0313.43018MR383002
  14. H. G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 188 (1979), 451-471 Zbl0447.43004MR599884
  15. H. G. Feichtinger, Banach convolution algebras of Wiener type, In (1983), 509-524 Zbl0528.43001MR751019
  16. H. G. Feichtinger, Generalized amalgams, with applications to Fourier transform, Canad. J. Math. 42 (1990), 395-409 Zbl0733.46014MR1062738
  17. H. G. Feichtinger, Modulation spaces on locally compact abelian groups, In (2003), 99-140, Chennai, India 
  18. H. G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I, J. Functional Anal. 86 (1989), 307-340 Zbl0691.46011MR1021139
  19. H. G. Feichtinger, K. Gröchenig, Gabor wavelets and the Heisenberg group: Gabor expansions and short time fourier transform from the group theoretical point of view, (1992), 359-398, Wavelets: A tutorial in theory and applications Zbl0849.43003MR1161258
  20. H. G. Feichtinger, K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Functional Anal. 146 (1997), 464-495 Zbl0887.46017MR1452000
  21. G. B. Folland, Harmonic Analysis in Phase Space, (1989), Princeton Univ. Press, Princeton, NJ Zbl0682.43001MR983366
  22. J. J. F. Fournier, J. Stewart, Amalgams of L p and l q , Bull. Amer. Math. Soc. (N.S.) 13 (1985), 1-21 Zbl0593.43005MR788385
  23. I. Gel’fand, D. Raikov, G. Shilov, Commutative normed rings, (1964), Chelsea Publishing Co., New York 
  24. K. Gröchenig, Foundations of time-frequency analysis, (2001), Birkhäuser Boston Inc., Boston, MA Zbl0966.42020MR1843717
  25. K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10 (2004), 105-132 Zbl1055.42018MR2054304
  26. K. Gröchenig, Composition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math. 98 (2006), 65-82 Zbl1148.47036MR2254480
  27. K. Gröchenig, Time-frequency analysis of Sjöstrand’s class, Revista Mat. Iberoam 22 (2006), 703-724 Zbl1127.35089
  28. K. Gröchenig, C. Heil, Modulation spaces and pseudodifferential operators, Integral Equations Operator Theory 34 (1999), 439-457 Zbl0936.35209MR1702232
  29. K. Gröchenig, C. Heil, Modulation spaces as symbol classes for pseudodifferential operators, (2003), 151-170, Wavelets and Their Applications 
  30. K. Gröchenig, M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), 1-18 Zbl1037.22012
  31. K. Gröchenig, S. Samarah, Non-linear approximation with local Fourier bases, Constr. Approx. 16 (2000), 317-331 Zbl0973.42025MR1759892
  32. E. Hernández, G. Weiss, A first course on wavelets, (1996), CRC Press, Boca Raton, FL Zbl0885.42018MR1408902
  33. L. Hörmander, The analysis of linear partial differential operators. III, 274 (1985), Springer-Verlag, Berlin Zbl0601.35001MR781536
  34. S. Jaffard, Propriétés des matrices “bien localisées” près de leur diagonale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 461-476 Zbl0722.15004
  35. N. Lerner, Y. Morimoto, A Wiener algebra for the Fefferman-Phong inequality, Sémin. Équ. Dériv. Partielles (2006), Seminaire: Equations aux Dérivées Partielles. 2005–2006 Zbl1122.35163MR2276082
  36. C. E. Rickart, General theory of Banach algebras, (1960), The University Series in Higher Mathematics. D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York Zbl0095.09702MR115101
  37. R. Rochberg, K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, (1998), 171-192, Gabor analysis and algorithms, Birkhäuser Boston, Boston, MA Zbl0890.42009MR1601103
  38. W. Rudin, Functional analysis, (1973), McGraw-Hill Book Co., New York Zbl0253.46001MR365062
  39. J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994), 185-192 Zbl0840.35130MR1266757
  40. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire sur les Équations aux Dérivées Partielles, 1994–1995 (1995), pages Exp. No. IV, 21 École Polytech, Palaiseau Zbl0880.35145MR1362552
  41. J. Sjöstrand, Pseudodifferential operators and weighted normed symbol spaces, Preprint (2007) Zbl1199.35410MR2414412
  42. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, (1993), Princeton Univ. Press, Princeton, NJ Zbl0821.42001MR1232192
  43. E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton Univ. Press, Princeton, NJ Zbl0232.42007MR304972
  44. J. Toft, Subalgebras to a Wiener type algebra of pseudo-differential operators, Ann. Inst. Fourier (Grenoble) 51 (2001), 1347-1383 Zbl1027.35168MR1860668
  45. J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal. 207 (2004), 399-429 Zbl1083.35148MR2032995
  46. J. Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II, Ann. Global Anal. Geom. 26 (2004), 73-106 Zbl1098.47045MR2054576
  47. J. Toft, Continuity and Schatten properties for pseudo-differential operators on modulation spaces, 172 (2007), 173-206, Oper. Theory Adv. Appl., Birkhäuser, Basel Zbl1133.35110MR2308511
  48. J. Ueberberg, Zur Spektralinvarianz von Algebren von Pseudodifferentialoperatoren in der L p -Theorie, Manuscripta Math. 61 (1988), 459-475 Zbl0674.47033MR952090

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