A Paley-Wiener theorem for the inverse Fourier transform on some homogeneous spaces

We formulate and prove a version of Paley-Wiener theorem for the inverse Fourier transform on non-compact Riemannian symmetric spaces and Heisenberg groups. The main ingredient in the proof is the Gutzmer's formula.Comment: 17 page

An analogue of Gutzmer's formula for Hermite expansions

We prove an analogue of Gutzmer's formula for Hermite expansions. As a consequence we obtain a new proof of a characterisation of the image of $L^2(\R^n)$ under the Hermite semigroup. We also obtain some new orthogonality relations for complexified Hermite functions.Comment: 15 page

Heat kernel transform for nilmanifolds associated to the Heisenberg group

We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.Comment: Revised version; to appear in Revista Mathematica Iberoamericana, 28

Mixed norm estimates for the Riesz transforms associated to Dunkl harmonic oscillators

In this paper we study weighted mixed norm estimates for Riesz transforms associated to Dunkl harmonic oscillators. The idea is to show that the required inequalities are equivalent to certain vector valued inequalities for operator defined in terms of Laguerre expansions. In certain cases the main result can be deduced from the corresponding result for Hermite Riesz transforms.Comment: 25 page

Hardy's inequality for fractional powers of the sublaplacian on the Heisenberg group

We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haagerup. The approach to prove the integral representations is via the language of semigroups. As a consequence of the Hardy inequalities we also obtain versions of Heisenberg uncertainty inequality for the fractional sublaplacian.Comment: 35 pages. Revised versio