A Beurling theorem for exponential solvable Lie groups (Q2791851)

For an exponential solvable Lie group \(G\) with non-trivial center, the authors show that a measurable function \(f\) on \(G\) satisfying NEWLINE\[NEWLINE\int_G\int_{\mathcal W}|f(g)|^2\|K_\xi^{1/2}\pi_\xi(f)\|^2_{HS} e^{2\|g\|\|\xi\|}\,dg\,d\xi<\inftyNEWLINE\]NEWLINE is 0 a.e. Here the integral over the cross section \(\mathcal W\) of coadjoint orbits is as in the Plancherel formula for exponential Lie groups.NEWLINENEWLINEThis result extends a theorem of Beurling [\textit{L. Hörmander}, Ark. Mat. 29, No. 2, 237--240 (1991; Zbl 0755.42009)] to this class of Lie groups.