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A Bochner type theorem for inductive limits of Gelfand pairs

Marouane Rabaoui (2008)

Annales de l’institut Fourier

In this article, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs ( G , K ) defined as inductive limits of increasing sequences of Gelfand pairs ( G ( n ) , K ( n ) ) n 1 . By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element ϕ of the set 𝒫 ( G ) of K -biinvariant continuous functions of positive type on G .

A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka, Yves Cornulier (2010)

Colloquium Mathematicae

Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation λ G / H of G on L²(G/H) has a spectral gap, that is, the restriction of λ G / H to the orthogonal complement of the constants in L²(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on n

E. K. Narayanan, S. Thangavelu (2006)

Annales de l’institut Fourier

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on n . If f ( z ) e 1 4 | z | 2 is a Schwartz class function we show that f is supported in a ball of radius B in n if and only if f × μ r ( z ) = 0 for r > B + | z | for all z n . This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When n = 1 we show that the two conditions f × μ r ( z ) = μ r × f ( z ) = 0 for r > B + | z | imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...

A weak type (1,1) estimate for a maximal operator on a group of isometries of a homogeneous tree

Michael G. Cowling, Stefano Meda, Alberto G. Setti (2010)

Colloquium Mathematicae

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.

A weighted Plancherel formula II. The case of the ball

Genkai Zhang (1992)

Studia Mathematica

The group SU(1,d) acts naturally on the Hilbert space L ² ( B d μ α ) ( α > - 1 ) , where B is the unit ball of d and d μ α the weighted measure ( 1 - | z | ² ) α d m ( z ) . It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

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