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

It is well known that (U(p,q),Hₙ) is a generalized Gelfand pair. Applying the associated spectral analysis, we prove a theorem of Wiener Tauberian type for the reduced Heisenberg group, which generalizes a known result for the case p = n, q = 0.

Let N be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra of dimension n. Let H be a subgroup of the automorphism group of N. Assume that H is a commutative, simply connected, connected Lie group with Lie algebra . Furthermore, assume that the linear adjoint action of on is diagonalizable with non-purely imaginary eigenvalues. Let $\tau =In{d}_H^{N⋊H}1$. We obtain an explicit direct integral decomposition for τ, including a description of the spectrum as a submanifold of (+)*, and a...

We define partial spectral integrals $S_R$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $log(2+L_{\alpha })f\in L²$, where $L_{\alpha }$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R}f\left(x\right)$ converges a.e. to f(x) as R → ∞.

Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_{R}f:={\int }_0^{R}d{E}_{\lambda }f,R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_0^{\infty }\lambda d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that $S_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log(2+ℒ)f\in L^2\left(G\right).$

We study the problem of $L^p$-boundedness ($1\<p\<\infty$) of operators of the form $m(L_1,\cdots ,L_n)$ for a commuting system of self-adjoint left-invariant differential operators $L_1,\cdots ,L_n$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_1,\cdots ,L_n$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.