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Anti-Wick quantization with symbols in $L^p$ spaces

Authors: Paolo Boggiatto and Elena Cordero
Journal: Proc. Amer. Math. Soc. 130 (2002), 2679-2685
MSC (2000): Primary 47G30, 35S05
Published electronically: February 4, 2002
MathSciNet review: 1900876
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Abstract: We give a classification of pseudo-differential operators with anti-Wick symbols belonging to $L^p$ spaces: if $p=1$ the corresponding operator belongs to trace classes; if $1\leq p\leq 2$ we get Hilbert-Schmidt operators; finally, if $p<\infty$, the operator is compact. This classification cannot be improved, as shown by some examples.

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  • 1. F.A. Berezin, Wick and Anti-Wick operator symbols, Math. Sb. 86(128), (1971), pp. 578-610; Math. USSR Sb. 15 (1971), pp. 577-606. MR 45:929
  • 2. B. Boggiatto, E. Buzano, L. Rodino, Global Hypoellipticity and Spectral Theory, Akademie Verlag, Berlin, Vol. 92, (1996). MR 97m:35040
  • 3. P. Boggiatto, L. Rodino, Partial Differential Equations of Multi-Quasi-Elliptic Type, Ann. Univ. Ferrara - Sez. VII - Sc. Mat. Suppl. Vol. XLV, pp. 275-291 (1999). CMP 2001:07
  • 4. A. Calderon, R. Vaillancourt, On the Boundedness of Pseudo-differential Operators, J. Math Soc. Japan, 23, (1971), pp.374-378. MR 44:2096
  • 5. A. Calderon, R. Vaillancourt, A Class of Bounded Pseudo-differential Operators, Proc. Nat. Acad. Sci. USA, 69 (1972) pp.1185-1187. MR 45:7532
  • 6. I. Daubechies, Ten Lectures on Wavelets, Capital City Press, CBMS-NSF Regional Conference Series in Appl. Math. (1992). MR 93e:42045
  • 7. J. Du, M.W. Wong, A Product Formula for Localization Operators, Bull. Korean Math. Soc. 37 (2000), No. 1, pp. 77-84. MR 2000m:47066
  • 8. G. B. Folland, Harmonic Analysis in Phase Space. Annals of Math. Studies, 122 (1989). Princeton University Press, Princeton, NJ. MR 92k:22017
  • 9. D. Robert, Autour de l'Approximation Semi-Classique, Birkhäuser-Verlag, Progress in Mathematics, Vol. 68 (1987). MR 89g:81016
  • 10. M. A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin (1987). MR 88c:47105
  • 11. B. Simon, The Weyl Transform and $L^p$ Functions on Phase Space, Proc.Amer.Math.Soc.4 (1992). MR 93b:81131
  • 12. E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, 32 Princeton University Press, (1971). MR 46:4102
  • 13. M.W. Wong, Weyl Transforms, Springer-Verlag, New York, Universitext, 1998. MR 2000c:47098

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Additional Information

Paolo Boggiatto
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Elena Cordero
Affiliation: Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Keywords: Anti-Wick, pseudo-differential operators, Hilbert-Schmidt, trace class
Received by editor(s): April 12, 2001
Published electronically: February 4, 2002
Communicated by: David Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

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