Positivity 8: 407–422, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Hypoelliptic Jacobi Convolution Operators on
J.J. BETANCOR, J.D. BETANCOR and J.M.R. MÉNDEZ
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 – La Laguna, Tenerife,
Islas Canarias, España. E-mail; email@example.com; firstname.lastname@example.org; email@example.com
(Received 20 September 2002; accepted 15 July 2003)
Abstract. In this paper we characterize the hypoellipticity of Jacobi convolution operators on
Schwartz distributions. In the proof of the main result of this paper the positivity of the convolution
structure for the inverse of the Jacobi transform plays an essential role. We also study hypoelliptic
convolution equations on Chébli-Trimèche hypergroups.
Mathematics Subject Classiﬁcation 2000: 46F12
Key words: Jacobi convolution equations, hypoelliptic, Schwartz distributions, Chébli-Trimèche
Harmonic analysis associated with the Jacobi operator
, t∈R, was ﬁrstly developed by M.
Flensted-Jensen and T.H. Koornwinder [10–12,15].
For every >−
and ∈ R, the Jacobi function
is deﬁned by
t∈0 and ∈ C
where =++1 and
denotes as usual the Gauss hypergeometric function.
As it is well known
is the even and C
-function on R which satisﬁes the
initial value problem
The Jacobi transform is deﬁned by