An uncertainty principle of Paley and Wiener on Euclidean motion group (Q1691990)

A classical result due to Paley and Wiener characterizes the existence of a non-zero function in \(L^2(\mathbb R),\) supported on a half line, in terms of the decay of its Fourier transform. In this paper the authors prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group \( M(n).\) They also relate this result to a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on \(M(n)\). The authors show: Let \(T_{r,\lambda } \) denote the unitary irreducible representation of \(M(n)\) associated to \(r>0,\) \( \lambda \) a unitary irreducible representation for \(\mathrm{SO}(n-1)\). Let \(f \) be a compactly supported continuous function on \(M(n)\) satisfying the estimate, \((\star),\) \( \| \hat f(T_{r,\lambda }) \|_{HS} \leq C_\lambda e^{-\theta (r)}\), for almost every \( r \in (0,\infty)\), where \(\theta \) is a non-negative locally integrable function on \([0, \infty)\). If \(I :=\int_0^\infty \frac{\theta (r)}{1+ r^2} dr =\infty\) then \(f = 0\) on \(M(n)\). Moreover, if \(\theta : [0, \infty) \rightarrow [0, \infty) \) is a non-decreasing function such that \(I\) is finite, then there exists a non-zero \(f \in C_c(M(n)) \)satisfying \((\star).\) The application to the Schrödinger equation is: Let \(f \in C_c(\mathbb R^n), u \in C(R: L^2(\mathbb R^n))\) be the solution of the equation \(u_t(x,t)=i\Delta u(x,t), u(x,0)=f(x), (x,t) \in \mathbb R^n\) and \(\theta\) be a non-negative locally integrable function on \([0, \infty)\). Assume that for some \(t_0 \not= 0\), \(| u (x, t_0) | \leq C e^{-\theta (x)}\), for almost every \(x \in \mathbb R^n\). If \(\int_0^\infty \frac{\theta(r)}{1 + r^2} dr = \infty \), then \( u = 0.\) The note contains an overview on its subject.