We prove that the Gelfand transform is a topological isomorphism between the space of polyradial Schwartz functions on the Heisenberg group and the space of Schwartz functions on the Heisenberg brush. We obtain analogous results for radial Schwartz functions on Heisenberg type groups.
Let H n be the (2n + 1)-dimensional Heisenberg group and K a compact group of automorphisms of H n such that (K H n , K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on H n and the space of Schwartz function on a closed subset of R s homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on H n .
Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of TV, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by
Let G be a simple Lie group of real rank one, with Iwasawa decomposition KA % N and Bruhat big cell NMA % N: Then the space G=MA % N may be (almost) identified with N and with K=M; and these identifications induce the (generalised) Cayley transform C : N-K=M: We show that C is a conformal map of Carnot-Caratheodory manifolds, and that composition with the Cayley transform, combined with multiplication by appropriate powers of the Jacobian, induces isomorphisms of Sobolev spaces H a ðNÞ and H a ðK=MÞ: We use this to construct uniformly bounded and slowly growing representations of G: r 2004 Elsevier Inc. All rights reserved.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.