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We study $¹-L^p$ boundedness of certain multiplier transforms associated to the special Hermite operator.

Let ℒ be the sublaplacian on the Heisenberg group Hⁿ. A recent result of Müller and Stein shows that the operator ${ℒ}^{-1/2}sin\surd ℒ$ is bounded on $L^p\left(Hⁿ\right)$ for all p satisfying |1/p - 1/2| < 1/(2n). In this paper we show that the same operator is bounded on $L^p$ in the bigger range |1/p - 1/2| < 1/(2n-1) if we consider only functions which are band limited in the central variable.

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 $r\&gt;B+\left|z\right|$ 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 $r\&gt;B+\left|z\right|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...