Subalgebras to a Wiener type algebra of pseudo-differential operators

Joachim Toft[1]

  • [1] Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 5, page 1347-1383
  • ISSN: 0373-0956

Abstract

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We study general continuity properties for an increasing family of Banach spaces S w p of classes for pseudo-differential symbols, where S w ∞ = S w was introduced by J. Sjöstrand in 1993. We prove that the operators in Op ( S w p ) are Schatten-von Neumann operators of order p on L 2 . We prove also that Op ( S w p ) Op ( S w r ) ⊂ Op ( S w r ) and S w p · S w q ⊂ S w r , provided 1 / p + 1 / q = 1 / r . If instead 1 / p + 1 / q = 1 + 1 / r , then S w p w * S w q ⊂ S w r . By modifying the definition of the S w p -spaces, one also obtains symbol classes related to the S ( m , g ) spaces.

How to cite

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Toft, Joachim. "Subalgebras to a Wiener type algebra of pseudo-differential operators." Annales de l’institut Fourier 51.5 (2001): 1347-1383. <http://eudml.org/doc/115950>.

@article{Toft2001,
abstract = {We study general continuity properties for an increasing family of Banach spaces $S^p_w$ of classes for pseudo-differential symbols, where $S^\infty _w=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in $\{\rm Op\}(S^p_w)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that $\{\rm Op\}(S^p_w)\{\rm Op\}(S^r_w)\subset \{\rm Op\}(S^r_w)$ and $S^p_w\cdot S^q_w\subset S^r_w$, provided $1/p + 1/q =1/r$. If instead $1/p +1/q = 1+1/r$, then $S^p_ww * S^q_w\subset S^r_w$. By modifying the definition of the $S^p_w$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.},
affiliation = {Blekinge Technical University, Department of Mathematics, IHN, 371-79 Karlskrona (Suède)},
author = {Toft, Joachim},
journal = {Annales de l’institut Fourier},
keywords = {pseudo-differential operators; Weyl calculus; Schatten-von Neumann classes; admissible functions; Hölder's inequality; Young's inequality; subalgebras; properties of pseudodifferential operators; symbols},
language = {eng},
number = {5},
pages = {1347-1383},
publisher = {Association des Annales de l'Institut Fourier},
title = {Subalgebras to a Wiener type algebra of pseudo-differential operators},
url = {http://eudml.org/doc/115950},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Toft, Joachim
TI - Subalgebras to a Wiener type algebra of pseudo-differential operators
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 5
SP - 1347
EP - 1383
AB - We study general continuity properties for an increasing family of Banach spaces $S^p_w$ of classes for pseudo-differential symbols, where $S^\infty _w=S_w$ was introduced by J. Sjöstrand in 1993. We prove that the operators in ${\rm Op}(S^p_w)$ are Schatten-von Neumann operators of order $p$ on $L^2$. We prove also that ${\rm Op}(S^p_w){\rm Op}(S^r_w)\subset {\rm Op}(S^r_w)$ and $S^p_w\cdot S^q_w\subset S^r_w$, provided $1/p + 1/q =1/r$. If instead $1/p +1/q = 1+1/r$, then $S^p_ww * S^q_w\subset S^r_w$. By modifying the definition of the $S^p_w$-spaces, one also obtains symbol classes related to the $S(m,g)$ spaces.
LA - eng
KW - pseudo-differential operators; Weyl calculus; Schatten-von Neumann classes; admissible functions; Hölder's inequality; Young's inequality; subalgebras; properties of pseudodifferential operators; symbols
UR - http://eudml.org/doc/115950
ER -

References

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  1. A. Boulkhemair, Remarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. L. 4 (1997), 53-67 Zbl0905.35103MR1432810
  2. J. Bergh, J. Löfström, Interpolation Spaces. An introduction, (1976), Springer-Verlag, Berlin-Heidelberg-New York Zbl0344.46071MR482275
  3. M. Dimassi, J. Sjöstrand, Spectral Asymptotics in the Semi-Classical Limit, vol. 268 (1999), Cambridge University Press, Cambridge, New York, Melbourne, Madrid Zbl0926.35002MR1735654
  4. L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. III (1985), Springer-Verlag, Berlin-Heidelberg-New York-Tokyo Zbl0601.35001MR404822
  5. E. H. Lieb, Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179-208 Zbl0726.42005MR1069246
  6. M. Reed, B. Simon, Methods of modern mathematical physics, (1979), Academic Press, London-New York Zbl0405.47007
  7. B. Simon, Trace ideals and their applications I, vol. 35 (1979), Cambridge University Press, Cambridge-London-New York-Melbourne Zbl0423.47001MR541149
  8. J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. L. 1 (1994), 185-192 Zbl0840.35130MR1266757
  9. J. Sjöstrand, Wiener type algebras of pseudodifferential operators, Séminaire Equations aux Dérivées Partielles, Ecole Polytechnique, 1994 n°IV (1995) Zbl0880.35145
  10. J. Toft, Continuity and Positivity Problems in Pseudo-Differential Calculus, (1996) 
  11. J. Toft, Regularizations, decompositions and lower bound problems in the Weyl calculus, Comm. Partial Differential Equations 7-8 (2000), 1201-1234 Zbl0963.35215
  12. J. Toft, Continuity properties in non-commutative convolution algebras with applications in pseudo-differential calculus, Bull. Sci. Math. 125 (2001) Zbl1002.43003MR1906240

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