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
Full Text:

References:

 [1] Agranovsky, M. L.; Rawat, Rama, Injectivity sets for the spherical means on the Heisenberg group, J. Fourier Anal. Appl., 5, 4, 363-372, (1999) · Zbl 0931.43007 [2] Bray, W. O., A spectral Paley-Wiener theorem, Monatsh. Math., 116, 1, 1-11, (1993) · Zbl 0803.43001 [3] Bray, W. O., Generalized spectral projections on symmetric spaces of non compact type: Paley-Wiener theorems, J. Funct. Anal., 135, 1, 206-232, (1996) · Zbl 0848.43009 [4] Epstein, C. L.; Kleiner, B., Spherical means in annular regions, Comm. Pure Appl. Math., 46, 3, 441-451, (1993) · Zbl 0841.31006 [5] Helgason, S., Groups and Geometric Analysis, (1984), Academic press, New York · Zbl 0543.58001 [6] Narayanan, E. K.; Thangavelu, S., Injectivity sets for the spherical means on the Heisenberg group, J. Math. Anal. Appl., 263, 2, 565-579, (2001) · Zbl 0995.43003 [7] Olver, F. W. J., Asymptotics and special functions, (1974), Academic press, New York · Zbl 0303.41035 [8] Rudin, W., Function theory in the unit ball of $$\mathbb{C}^n, 241, (1980),$$ Springer-Verlag, New York - Berlin · Zbl 0495.32001 [9] Sajith, G.; Thangavelu, S., On the injectivity of twisted spherical means on $$\mathbb{C}^n,$$ Israel J. Math., 122, 79-92, (2) · Zbl 0986.43001 [10] Strichartz, R. S., Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal., 87, 51-148, (1989) · Zbl 0694.43008 [11] Szego, G., Orthogonal polynomials, 23, (1967), Amer. Math. Soc., Providence, R. I. · JFM 61.0386.03 [12] Thangavelu, S., Lectures on Hermite and Laguerre expansions, 42, (1993), Princeton University Press, Princeton, NJ · Zbl 0791.41030 [13] Thangavelu, S., Harmonic analysis on the Heisenberg group, 159, (1998), Birkhäuser Boston, Boston, MA · Zbl 0892.43001 [14] Thangavelu, S., An introduction to the uncertainty principle, 217, (2004), Birkhäuser Boston, Boston, MA · Zbl 1188.43010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.