Injectivity sets for spherical means on the Heisenberg group. (English) Zbl 0931.43007

Let \( \mu _{r}\), \(r>0\), be the normalized surface measure on the sphere \(\{w\in {\mathbb C^n} : |w|=r\}\) in \({\mathbb C^n}\). For \(f\in C({\mathbb C^n})\), the twisted spherical means of \(f\) are defined as follows: \[ f\times \mu _r(z)=\int_{|w|=r} f(z-w)e^{i {1\over 2} \operatorname{Im} (z\cdot \overline w)} d\mu_r(w),\quad z\in {\mathbb C}^n. \] A set \(S\subset {\mathbb C}^n\) is said to be a set of injectivity for the twisted spherical means in a subclass \(\mathcal C\) of the continuous functions on \({\mathbb C}^n\) if \(f\in \mathcal C\) and \(f\times \mu_r(z)=0\), for all \(r\geq 0\) and \(z\in S\), imply that \(f=0\). The authors consider functions that satisfy \(f(z)e^{({1\over {4}}+\varepsilon)|z|^2}\in L^p({\mathbb C}^n)\) for some \(\varepsilon >0\) and \(1\leq p \leq \infty \). They show that in this class of functions, the boundary of a bounded domain in \({\mathbb C}^n\) is a set of injectivity for the twisted spherical means. As a consequence some results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform are given.


43A80 Analysis on other specific Lie groups
44A12 Radon transform
Full Text: DOI EuDML


[1] Agranovsky, M.L., Berenstein, C., Chang, D.C., and Pascuas, D. (1994). Injectivity of the Pompeiu transform in the Heisenberg group,J. Anal. Math.,63, 131-173. · Zbl 0808.43002
[2] Agranovsky, M.L., Berenstein, C., and Kuchment, P. (1998). Approximation by spherical waves inL p-spaces,J. Geom. Anal.,6, 365-383. · Zbl 0898.44003
[3] Agranovsky, M.L. and Quinto, E.T. (1996). Injectivity sets for the Radon transform over circles and complete systems of radial functions,J. Funct Anal.,139, 383-414. · Zbl 0860.44002
[4] Agranovsky, M.L., Volchkov, V.V., and Zalcman, L.A. (1999). Conical uniqueness sets for the spherical Radon transform,Bull. London Math. Soc.,31, to appear. · Zbl 0935.44001
[5] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G. (1953).Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York. · Zbl 0051.30303
[6] Olver, F.M.J. (1974).Introduction to Asymptotics and Special Functions, Academic Press, New York. · Zbl 0308.41023
[7] Rawat, R. and Sitaram, A. (preprint). Injectivity sets for spherical means on ?n and on symmetric spaces. · Zbl 0952.43001
[8] Thangavelu, S. (1991). Spherical means on the Heisenberg group and a restriction theorem for the symplectic Fourier transform,Revist. Mat. Ibero.,7, 135-155. · Zbl 0785.43003
[9] Thangavelu, S. (1993).Lectures on Hermite and Laguerre Expansions, Math Notes, Princeton University Press, Princeton, NJ. · Zbl 0791.41030
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