Spherical means and CR functions on the Heisenberg group
The injectivity of the spherical mean value operator is studied. The main concern of this paper is the case of the spherical mean on the Heisenberg group. Let \(\mu_ r\) denote the normalized surface measure on \(\{(z, 0): | z| = r\}\) in the Heisenberg group \(\mathbb{H}^ n = \mathbb{C}^ n \times \mathbb{R}\). It is proved that the spherical mean value operator \(f \to f * \mu_ r\) on the Heisenberg group is injective when \(f \in L^ p(\mathbb{H}^ n)\), \(1 \leq p < \infty\). When \(f(z, \cdot) \in L^ p (\mathbb{R})\), \(1 \leq p \leq 2\), the same result is proved under much weaker conditions such that the partial Fourier transform \(f^ \lambda (z) = \int^ \infty_{-\infty} e^{i \lambda t} f(z,t) dt\) is of tempered growth on \(\mathbb{C}^ n\) for a.e. \(\lambda\). Moreover, some results regarding CR functions on the Heisenberg group are proved. Those results are extensions of recent ones obtained by \textit{M. Agranovsky, C. Berenstein, D. Chang} and \textit{D. Pascuas} [J. Anal. Math. 57, 282-296 (1991; Zbl 0773.32012)].
scientific article; zbMATH DE number 569423
scientific article