Coxeter system of lines and planes are sets of injectivity for the twisted spherical means (Q2253257)

Injectivity sets for the twisted spherical means (TSM for short) are investigated. It is proved, among others, that: (a) any line passing through the origin is a set of injectivity for the TSM for functions \(f\in L^2(\mathbb C)\) such that every spectral projector \(\big(\exp\big(\frac14|\cdot|^2\big)f\big)\times \varphi_k\), \(k\in\mathbb N_0\), is a polynomial; (b) any Coxeter system of even number of lines is a set of injectivity for the TSM for any \(f\in L^p(\mathbb C)\), \(1\leq p\leq2\). On the other hand, in the case \(n\geq2\), it is proved that a certain Coxeter system of even number of hyperplanes is a set of injectivity for the TSM for any \(f\in L^p(\mathbb C^n)\), \(1\leq p\leq2\).