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

Narayanan, E. K.; Thangavelu, S.

Ann. Inst. Fourier 56, No. 2, 459-473 (2006).

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 \(z \in {\mathbb C}\).

Reviewer: Koichi Saka (Akita)

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

Keywords:

spectral decomposition; Paley-Wiener theorem; Heisenberg group; support theorem; spherical mean

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