A Beurling-Hörmander theorem associated with the Riemann-Liouville operator (Q539030)

The uncertainty principle is not only of great importance in various subjects but can be presented in many different forms and settings. One of the most general forms for the usual Fourier transform \(\widehat f\) of \(f\in L^2(\mathbb R^d)\) is the classical Beurling-Hörmander theorem which states that if for \(N\geq0\) \[ \int_{\mathbb R^d}\int_{\mathbb R^d}\frac{|f(x)|\,|\widehat f(y)|e^{|x|\,|y|}} {(1+|x|+|y|)^N}\,dx\,dy<\infty, \] then \(f(x)=P(x)e^{-a\langle Ax,x\rangle},\) \(a>0,\) where \(A\) is a real positive definite symmetric matrix and \(P\) is a polynomial of degree \(<(N-d)/2\). In particular, \(f=0\) when \(N\leq d\). In the paper under review, the author extends this result to a generalized Fourier transform associated with the Riemann-Liouville operator. As a corollary of his main result, the author derives some other versions of the uncertainty principle.