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We compute the “norm” of irreducible uniformly bounded representations of SL(2,R). We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in its similarity class of uniformly bounded representations.

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2, R) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal

We compute the “norm” of irreducible uniformly bounded representations of SL(2,R). We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in its similarity class of uniformly bounded representations.

We estimate the norms of many matrix coefficients of irreducible uniformly bounded representations of SL(2,ℝ) as completely bounded multipliers of the Fourier algebra. Our results suggest that the known inequality relating the uniformly bounded norm of a representation and the completely bounded norm of its coefficients may not be optimal.

We compute the “norm” of irreducible uniformly bounded representations of SL(2, R). We show that the Kunze–Stein version of the uniformly bounded representations has minimal norm in the similarity class of uniformly bounded representations

We prove several Paley–Wiener-type theorems related to the spherical transform on the Gelfand pair $$\big ({H_n}\rtimes {\text {U}(n)},{\text {U}(n)}\big )$$(Hn⋊U(n),U(n)), where $${H_n}$$Hn is the $$2n+1$$2n+1-dimensional Heisenberg group. Adopting the standard realization of the Gelfand spectrum as the Heisenberg fan in $$\mathbb {R}^2$$R2, we prove that spherical transforms of $${\text {U}(n)}$$U(n)-invariant functions and distributions with compact support in $${H_n}$$Hn admit unique entire extensions to $$\mathbb {C}^2$$C2, and we find real-variable characterizations of such transforms. Next, we characterize the inverse spherical transforms of compactly supported functions and distributions on the fan, giving analogous characterizations.

We prove that Huygens’ principle and the principle of equipartition of energy hold for the modified wave equation on odd dimensional Damek–Ricci spaces. We also prove a Paley–Wiener type theorem for the inverse of the Helgason Fourier transform on Damek–Ricci spaces.

Let Hnbe the (2n+1)-dimensional Heisenberg group and Ka compact group of automorphisms of Hnsuch that (K⋉Hn,K) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on Hnand the space of Schwartz function on a closed subset of Rshomeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.

We prove that a Damek–Ricci space is symmetric if and only if the geodesic inversion preserves the set of horocycles.

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.

A distribution on a Heisenberg type group of homogeneous dimension Q a biradial kernel of type _it coincides with a biradial function, homogeneous of degree _ a - Q, and smooth away from the identity. We prove that a distribution is a biradial kernel of type a_, 0 _< a _ < Q, if and only if its Gelfand transform, defined on the Heisenberg fan, extends to a smooth even function on the upper half plane, homogeneous of degree -a/2. A similar result holds for radial kernels on the Heisenberg group.

The generalised Cayley transform C from an Iwasawa N-group into the corresponding real unit sphere S induces isomorphisms between suitable Sobolev spaces H_(S) and H_(N). We study the differential of C, and we obtain a criterion for a function to be in H_(S).

Let s the geodesic inversion on a Heisenberg type group N . We prove that s induced a bounded operator on suitable Sobolev spaces if and only if N is an Iwasawa N-group.

Let G be a simple Lie group of real rank one, with Iwasawa decomposition KAN and Bruhat big cell NMAN. Then the space G/MAN may be (almost) identified with N and with K/M; and these identifications induce the (generalised) Cayley transform C from N into 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 on N and K/M. We use this to construct bounded and slowly growing representations of G.

A harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups

Let N be an H-type group of homogeneous dimension e. We study the space of biradial Schwartz functions on N by means of the Gelfand transform. This enables us to characterize the class of biradial homogeneous distributions on N of degree alpha, with 0 less than or equal to alpha < Q, which are C-infinity away from the identity, via the Gelfand transform

Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, 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 E. Damek and F. Ricci

Let n = v + z be an H-type group, and let n + a be the harmonic semidirect product of n with a similar or equal to R. Let N A be the corresponding simply connected Lie group. If dim v = m and dim z = k, denote Q = m/2 + k. We prove that the spherical Fourier transform is a topological isomorphism between the p-Schwartz space L(p)(N A)(#), (0 < p less than or equal to 2), and the space of holomorphic rapidly decreasing functions on the strip {s is an element of C:\Re(s)\ < epsilon Q/2} with epsilon = 2/p - 1