Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hermite distributions
associated to the group $O(p,q)$


Author: Gerald B. Folland
Journal: Proc. Amer. Math. Soc. 126 (1998), 1751-1763
MSC (1991): Primary 33E30; Secondary 33C15, 35C05
DOI: https://doi.org/10.1090/S0002-9939-98-04331-7
MathSciNet review: 1451801
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We calculate the tempered $O(p,q)$-invariant eigendistributions of the $O(p,q)$-invariant Hermite operator

\begin{equation*}-{\textstyle {\frac{1}{2}}}(\Delta _{x}- \Delta _{y}) +{\textstyle {\frac{1}{2}}}(|x|^{2}-|y|^{2})\qquad (x\in \mathbb{R}^{p},\ y\in \mathbb{R}^{q}).\end{equation*}

They are singular on the cone $|x|=|y|$ and are given elsewhere in terms of confluent hypergeometric functions.


References [Enhancements On Off] (What's this?)

  • 1. The Bateman Manuscript Project (A. Erdélyi, director), Higher Transcendental Functions, vol. I, McGraw-Hill, New York, 1953. MR 15:419i
  • 2. G. de Rham, Sur la division de formes et de courants par une forme linéaire, Comm. Math. Helv. 28 (1954), 346-352. MR 16:402d
  • 3. Georges de Rham, Solution élémentaire d’opérateurs différentiels du second ordre, Ann. Inst. Fourier. Grenoble 8 (1958), 337–366 (French). MR 0117437
  • 4. Roger Howe and Eng-Chye Tan, Nonabelian harmonic analysis, Universitext, Springer-Verlag, New York, 1992. Applications of 𝑆𝐿(2,𝑅). MR 1151617
  • 5. P. D. Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz, Comm. Math. Helv. 28 (1954), 225-269. MR 16:255c
  • 6. A. Tengstrand, Distributions invariant under an orthogonal group of arbitrary signature, Math. Scand. 8 (1960), 201–218. MR 0126154, https://doi.org/10.7146/math.scand.a-10610

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 33E30, 33C15, 35C05

Retrieve articles in all journals with MSC (1991): 33E30, 33C15, 35C05


Additional Information

Gerald B. Folland
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195-4350
Email: folland@math.washington.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04331-7
Received by editor(s): December 5, 1996
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society