Injectivity sets for spherical means on \(\mathbb{R}^n\) and on symmetric spaces
A subset \(\Gamma\) of \(\mathbb{R}^n\) is called an NA-set if the only real analytic function defined in a neighborhood of \(\Gamma\) which vanishes on \(\Gamma\) is the zero function. A complex Radon measure \(\mu\) in \(\mathbb{R}^n\) is said to be of at most exponential-quadratic growth if for some constants \(C\), \(\alpha\) and any \(r>0\) one has \(|\mu|(B(0, r))\leq Ce^{\alpha r^2}\). The main result of this paper is the following uniqueness theorem: if \(\Gamma\) is an NA-set and \(\mu\) is a measure of at most exponential growth such that \(\mu(B(x, r))= 0\) for all \(x\in\Gamma\) and all \(r>0\), then \(\mu= 0\). This result compliments work on spherical means which can be found in, for example [\textit{M. L. Agranovsky} and \textit{E. T. Quinto}, J. Funct. Anal. 139, 383-414 (1996; Zbl 0860.44002)] and [\textit{M. L. Agranovsky}, \textit{C. A. Berenstein} and \textit{P. Kuchment}, J. Geom. Anal. 6, 365-383 (1998; Zbl 0898.44003)].
scientific article; zbMATH DE number 1445616