## A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on $$\mathbb C^n$$.(English)Zbl 1089.43006

The authors prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on $${\mathbb C}^n$$.
Let $${\mathbb H}^n$$ be the Heisenberg group which as a manifold is $${\mathbb C}^n \times {\mathbb R}$$ with the group law $(z, t)(w, s)=(z+w, t+s+\frac{1}{2}\text{ Im} (z\cdot {\bar w})).$ The spectral resolution of the sub-Laplacian $${\mathcal L}$$ on $${\mathbb H}^n$$ is written as $f(z, t) =(2\pi)^{-n-1}\sum^{\infty}_{k=0}\int^{\infty}_{-\infty} f*e^{\lambda}_{k}(z, t)|\lambda|^n d\lambda$ where the functions $$e^{\lambda}_{k}$$ are eigenfunctions of $${\mathcal L}$$ with eigenvalue $$(2k+n)|\lambda|$$ given by $e^{\lambda}_{k}(z, t) = e^{i\lambda t}\phi^{\lambda}_{k}(z)= e^{i\lambda t} \phi_{k}(\sqrt{|\lambda|}z)$ with the Laguerre functions $$\phi_{k}(z)=L^{n-1}_{k}(\frac{1}{2}|z|^2) e^{-\frac{1}{4}|z|^2}$$ of type $$(n-1)$$.
Let $$\Delta_{+}\psi (k) = \psi (k+1)- \psi (k), \;\;\Delta_{-}\psi (k) = \psi(k) - \psi(k-1)$$ be the forward and backward finite difference operators and let $$\Delta \psi (k) = k \Delta_{+} \Delta_{-}\psi(k) + n \Delta_{+}\psi(k)$$.
For each $$j \geq 0$$ let $$l^2_{j}$$ stand for all sequences $$\psi = (\psi(k))$$ for which $\| \psi\| _{2,j}=(\sum^{\infty}_{k=0}|\Delta^{j}\psi(k)|^2)^{1/2} < \infty.$ Let $$L^{p,q}({\mathbb H}^n), \;\;1 \leq p < \infty, \;1 \leq q \leq 2,$$ consist of functions for which $\| f\| _{p,q}=(\int_{{\mathbb C}^{n}}(\int_{{\mathbb R}}|f(z, t)|^{q} dt )^{p/q} dz )^{1/p} < \infty$ and let $$f^{\lambda}(z)$$ be the partial inverse Fourier transform of $$f(z,t)$$ in the $$t$$-variable.
Main theorem. Let $$f \in L^{p,q}({\mathbb H}^n), \;\;1 \leq p < \infty, \;1 \leq q \leq 2,$$ be such that $$f^{\lambda}(z)e^{\frac{1}{4}|\lambda| |z|^2}$$ is a Schwartz class function on $${\mathbb C}^n$$ for every $$\lambda \in {\mathbb R}.$$ Then $$f$$ is supported in $$\{ (z,t): |z| \leq B, \;t \in {\mathbb R} \}$$ if and only if for every $$j \geq 0$$ the sequence $$\psi_{\lambda} =(\psi_{\lambda}(k))$$, where $$\psi_{\lambda}=f*e^{\lambda}_{k} (z,t)$$, belongs to $$l^{2}_{j}$$ and $\| \psi_{\lambda}\| _{2,j} \leq C(\frac{1}{2}|\lambda|)^{j}(B+|z|)^{2j}.$ Let $$\mu_{r}$$ be the normalized surface measure on the sphere $$S_{r}=\{ z \in {\mathbb C}^n: |z|=r \}$$.
Define $f\times \mu_{r}(z)=\int_{|w|=r}f(z-w)e^{\frac{i}{2}\text{ Im} (z\cdot {\bar w})} d\mu_{r}(w).$ Support theorem. (a) Let $$f$$ be a function on $${\mathbb C}^n$$ such that $$f(z) e^{\frac{1}{4}|z|^2}$$ is in the Schwartz class. Then $$f$$ is supported in $$|z| \leq B$$ if and only if $$f \times \mu_{r}(z)=0$$ for $$r>B+|z|$$ for every $$z \in {\mathbb C}^n$$.
(b) Let $$f$$ be a locally integrable function on $${\mathbb C}$$ such that $$|f(z)| \leq C e^{\frac{1}{4}(1-\epsilon)|z|^2}$$ for some $$\epsilon > 0$$. Then $$f$$ is supported in $$|z| \leq B$$ if and only if $$f \times \mu_{r}(z) = 0$$ for $$r > B+|z|$$ for every $$z \in {\mathbb C}$$.

### MSC:

 43A80 Analysis on other specific Lie groups 22E30 Analysis on real and complex Lie groups 44A35 Convolution as an integral transform 53C65 Integral geometry
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