Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type

Abstract

We will show that an uniform treatment yields Wiener-Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/K . The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze-Stein phenomenon for noncompact semisimple Lie groups.

Spectral analysis for radial sections of some homogeneous vector bundles on certain noncompact Riemannian symmetric spaces

“…Except for the pair (L 1 , L ∞ ), these questions do not have any euclidean counterpart. These, vis-a-vis the Wiener-Tauberian theorems for various Lorentz spaces, are intrinsic to the structure of semisimple Lie groups and are rooted in the Kunze-Stein phenomenon (see [9,12,11,27].) First let us consider the case of the commutative Banach algebra L p,1 τ (G), 1 ≤ p < 2.…”

“…By Theorem 4.3 there exist f, g ∈ C p2 τ (G) such that f = F and g = G. It is clear that F does not vanish anywhere on S p . We recall that C p2 τ (G) is a dense subspace of L p,1 τ (G) (see [27]). We assume that…”

“…In this case a slightly stronger result is true: φ λ is in L ∞ (G) if and only if λ ∈ S 1 . A more detailed account on the estimates of φ λ can be found in [28,27] (see also Section 7).…”

An analogue of Krein’s theorem for semisimple Lie groups

“…Also, for r > 1, where 1/r + 1/r = 1 [Pusti et al 2011]. Now, from Theorem 3.3, a K -positive definite function can be expressed as…”

Wiener Tauberian theorem for rank one semisimple Lie groups and for hypergeometric transforms

“…With the extended strip condition the theorem above has been extended to the full group (see ), on rank one symmetric spaces (see ). See also , , , , for furthur reference. Y. Ben Natan, Y. Benyamini, H. Hedenmalm and Y. Weit (in , ) proved a genuine analogue of the Wiener Tauberian theorem without the extended strip condition on in the K ‐biinvariant setting.…”