A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform

Bent Ørsted[1]; Jorge Vargas[2]

  • [1] Ny Munkegade Department of Mathematics 8000 Aarhus C. (Danemark)
  • [2] Universidad Nacional de Córdoba FAMAF-CIEM Ciudad Universitaria 5000 Córdoba (Argentine)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 3, page 693-702
  • ISSN: 0373-0956

Abstract

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In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the ( L 2 , L 2 ) -continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is ( L p , L p ) -continuous for 1 ≤ p ≤ ∞ .

How to cite

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Ørsted, Bent, and Vargas, Jorge. "A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform." Annales de l’institut Fourier 57.3 (2007): 693-702. <http://eudml.org/doc/10237>.

@article{Ørsted2007,
abstract = {In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $(L^p, L^p)$-continuous for $ 1 \le p \le \infty .$},
affiliation = {Ny Munkegade Department of Mathematics 8000 Aarhus C. (Danemark); Universidad Nacional de Córdoba FAMAF-CIEM Ciudad Universitaria 5000 Córdoba (Argentine)},
author = {Ørsted, Bent, Vargas, Jorge},
journal = {Annales de l’institut Fourier},
keywords = {Discrete Series representations; branching laws; invariant elliptic operators; Berezin transform; unitary representation; Casimir operator; semisimple Lie group},
language = {eng},
number = {3},
pages = {693-702},
publisher = {Association des Annales de l’institut Fourier},
title = {A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform},
url = {http://eudml.org/doc/10237},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Ørsted, Bent
AU - Vargas, Jorge
TI - A Cauchy Problem for Elliptic Invariant Differential Operators and Continuity of a generalized Berezin transform
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 3
SP - 693
EP - 702
AB - In this note, we generalize the results in our previous paper on the Casimir operator and Berezin transform, by showing the $(L^2,L^2)$-continuity of a generalized Berezin transform associated with a branching problem for a class of unitary representations defined by invariant elliptic operators; we also show, that under suitable general conditions, this generalized Berezin transform is $(L^p, L^p)$-continuous for $ 1 \le p \le \infty .$
LA - eng
KW - Discrete Series representations; branching laws; invariant elliptic operators; Berezin transform; unitary representation; Casimir operator; semisimple Lie group
UR - http://eudml.org/doc/10237
ER -

References

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  8. Karl-Hermann Neeb, Holomorphy and Convexity in Lie Theory, 28 (2000), Walter de Gruyter Zbl0936.22001MR1740617
  9. B. Orsted, J. Vargas, Restriction of square integrable representations: Discrete Spectrum, Duke Math. Journal 123 (2004), 609-633 Zbl1056.22008MR2068970
  10. S. Treves, Topological vector spaces, distributions and kernels, (1967), Academic Press, New York Zbl0171.10402MR225131
  11. P. Trombi, V. Varadarajan, Asymptotic behavior of eigenfunctions on a semisimple Lie group: The Discrete Spectrum, Acta Math. 129 (1972), 237-280 Zbl0244.43006MR393349
  12. N. Wallach, J. Wolf, Completeness of Poincare Series for Automorphic Forms Associated to the Integrable Discrete Series, Representation Theory of reductive groups 40 (1983), 265-281, P. Trombi (editor) Zbl0566.22014MR733818

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