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On Paley-Wiener theorems for the Heisenberg group. (English) Zbl 0793.43006
Two theorems of Paley-Wiener type are proved for the Heisenberg group. The first one is for the group Fourier transform which is the analogue of the classical Paley-Wiener theorem. The second one is for the spectral projection associated to the sub-Laplacian on the Heisenberg group.
Let \(H^ n = \mathbb{C}^ n \times \mathbb{R}\) be the Heisenberg group and \(L^{(p,1)}(H^ n)\) be the space consisting of all functions \(f\) on \(H^ n\) for which \[ \| f\|_{p,1} =\Bigl(\int_{\mathbb{C}^ n}\bigl(\int^ \infty_{-\infty} | f(z,t)| dt\bigr)^ p dz\Bigr)^{1/p} < \infty. \] For each nonzero real number \(\lambda\), the operators \(U_{\lambda}(z,t)\) are the infinite dimensional irreducible unitary representations of \(H^ n\) and we set \(U_ \lambda(z,t) = e^{i \lambda t} V_ \lambda(z)\), \(z\in \mathbb{C}^ n\), \(t \in \mathbb{R}\). For a function \(f\) in \(L^ 1(H^ n)\), its Fourier transform \(\widehat{f}(\lambda)\) is defined to be \[ \widehat{f}(\lambda) = \int_{H^ n} f(z,t)e^{i \lambda t} V_ \lambda(z)dz dt. \] We define an operator valued function \(F_ \lambda(\xi',\xi'')\) on \(\mathbb{R}^ n \times \mathbb{R}^ n\) by \[ F_ \lambda(\xi',\xi'') = V_ \lambda(\xi' + i\xi'')\widehat{f}(\lambda)V_ \lambda (-\xi' - i \xi''). \]
The first theorem: Assume \(1 \leq p \leq 2\) and \(f \in L^{(p,1)}(H^ n)\).
(a) \(f\) is supported in \(| z| \leq B\) if and only if \(F(\xi',\xi'')\) extends to an entire function of \((\zeta',\zeta'')\) in \(\mathbb{C}^ n \times \mathbb{C}^ n\) satisfying the estimate \(\| F_ \lambda(\zeta',\zeta'')\| \leq C| \lambda|^{- n/p'}e^{B| \lambda| | \text{Im }\zeta|}\) where \(\zeta = (\zeta',\zeta'')\).
(b) \(f\) is supported in \(| x| \leq B\) and \(| y| \leq B\) if and only if \(F_ \lambda(\zeta',0)\) and \(F_ \lambda(0,\zeta'')\) extend to \(\mathbb{C}^ n\) as entire functions satisfying the estimates \[ \| F_ \lambda(\zeta',0)\| \leq C| \lambda|^{-n/p'} e^{B| \lambda| | \text{Im }\zeta'|}\qquad \| F_ \lambda(0,\zeta'')\| \leq C| \lambda|^{-n/p'} e^{B| \lambda| | \text{Im }\zeta''|}. \] We define a second order finite difference operator \(E\) by \(E \psi(k) = k\Delta_ + \Delta_ - \psi(k) + n\Delta_ + \psi(k)\) for \(k = 0,1,2,\dots\), where \(\Delta_ +\) and \(\Delta_ -\) stand for the forward and backward difference operators given by \(\Delta_ +\psi(k) = \psi(k+1) - \psi(k)\), \(\Delta_ -\psi(k) = \psi(k) - \psi(k-1)\). For each \(j = 0,1,2,\dots\), \(E^ j\) is defined inductively.
The second theorem: Let \(f\) be a function in \(C^ \infty(H^ n)\) supported in \(| z| \leq B\). Then it holds that there is a constant \(C > 0\) such that \[ \Bigl(\sum_{k = 0}| E^ j \bigl({k!(n-1)!\over (k+n-1)!} P_{k,\lambda} f(z,t)|^ 2 {(k+n- 1)!\over k!(n-1)!}\bigr)\Bigr)^{1/2} \leq C 2^{-j}(B + | z|)^{2j} | \lambda |^{-n+j} \] for each \(j = 0,1,2,\dots,\) where the operators \(P_{k,\lambda}\) are the spectral projection operators associated to \(i(\partial/\partial t)\) and the Heisenberg Laplacian in \(H^ n\). Conversely, if \(f\) is radial and the above estimates hold for each \(j\) then \(f\) is supported in \(| z| \leq B\).
Reviewer: K.Saka (Akita)

43A80 Analysis on other specific Lie groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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