Solvability of invariant sublaplacians on spheres and group contractions

Fulvio Ricci; Jérémie Unterberger

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2001)

  • Volume: 12, Issue: 1, page 27-42
  • ISSN: 1120-6330

Abstract

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In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians L α on the spheres S 2 ⁢ n + 1 ≃ U ⁢ n + 1 / U ⁢ n . In the second part, we introduce a larger family of left-invariant sublaplacians L α , β on S 3 ≃ S ⁢ U ⁢ 2 and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.

How to cite

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Ricci, Fulvio, and Unterberger, Jérémie. "Solvability of invariant sublaplacians on spheres and group contractions." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 12.1 (2001): 27-42. <http://eudml.org/doc/252417>.

@article{Ricci2001,
abstract = {In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians $\mathcal\{L\}_\{\alpha\}$ on the spheres $S^\{2n+1\} \simeq U(n+1)/U(n)$. In the second part, we introduce a larger family of left-invariant sublaplacians $\mathcal\{L\}_\{\alpha,\beta\}$ on $S^\{3\} \simeq SU(2)$ and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.},
author = {Ricci, Fulvio, Unterberger, Jérémie},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Local solvability; Hypoellipticity; Invariant differential operators; Lie group contractions; local solvability; hypoellipticity; invariant differential operators},
language = {eng},
month = {3},
number = {1},
pages = {27-42},
publisher = {Accademia Nazionale dei Lincei},
title = {Solvability of invariant sublaplacians on spheres and group contractions},
url = {http://eudml.org/doc/252417},
volume = {12},
year = {2001},
}

TY - JOUR
AU - Ricci, Fulvio
AU - Unterberger, Jérémie
TI - Solvability of invariant sublaplacians on spheres and group contractions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2001/3//
PB - Accademia Nazionale dei Lincei
VL - 12
IS - 1
SP - 27
EP - 42
AB - In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians $\mathcal{L}_{\alpha}$ on the spheres $S^{2n+1} \simeq U(n+1)/U(n)$. In the second part, we introduce a larger family of left-invariant sublaplacians $\mathcal{L}_{\alpha,\beta}$ on $S^{3} \simeq SU(2)$ and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.
LA - eng
KW - Local solvability; Hypoellipticity; Invariant differential operators; Lie group contractions; local solvability; hypoellipticity; invariant differential operators
UR - http://eudml.org/doc/252417
ER -

References

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  1. Boggess, A., C ⁢ R Manifolds and the Tangential Cauchy-Riemann Complex. Studies in Advanced Mathematics, CRC Press, 1991. Zbl0760.32001MR1211412
  2. Helgason, S., Groups and geometric analysis. Academic Press, 1984. Zbl0543.58001MR754767
  3. Hörmander, L., A Class of Hypoelliptic Pseudodifferential Operators with Double Characteristics. Math. Ann., vol. 217, 1975, 165-188. Zbl0306.35032MR377603
  4. Hua, L. K., Harmonic analysis of Functions of Several complex variables in the classical domains. Transl. Math. Monog., vol. 6, A.M.S., Providence, RI1963. Zbl0112.07402MR171936
  5. Korányi, A., The Poisson integral for generalized half-planes and bounded symmetric domains. Ann. Math., 82, 1965. Zbl0138.06601MR200478
  6. Nirenberg, L. - Trèves, F., On Local Solvability of Linear Partial Differential Equations. Part I : Necessary Conditions. Vol. 23, 1970, 1-38. Zbl0191.39103MR264470
  7. Ricci, F., A Contraction of S ⁢ U ⁢ 2 to the Heisenberg group. Mh. Mat., 101, 1986, 211-225. Zbl0588.43007MR847376DOI10.1007/BF01301660
  8. Stein, E.M., Harmonic analysis. Princeton University Press, 1983. Zbl0821.42001
  9. Taylor, M.E., Noncommutative harmonic analysis. AMS, Mathematical surveys and monographs, 1986. Zbl0604.43001MR852988
  10. Ja, N. - Klimyk, A. U., Representation of Lie Groups and Special Functions. Vol. 2, Kluwer Academic Publishers, 1993. Zbl0809.22001MR1220225

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