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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)

##### MSC:
 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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