On the Cowling-Price theorem for \(SU(1,1)\)
A generalization of a theorem of Hardy by \textit{M. G. Cowling} and \textit{J. F. Price} [Generalizations of Heisenberg's inequality, Lecture Notes Math. 992, 443-449 (Berlin 1983; Zbl 0516.43002)] asserts that for numbers \(p,q\) both greater or equal to 1 and one of which is finite, if a measurable function \(f\) on \(R\) is such that \(\exp\{ax^2\}f(x)\) belongs to \(L^p\) and \(\exp \{by^2\} f^\wedge(y)\) belongs to \(L^q\) for \(ab\geq 1/4\), then in fact \(f\) is 0 almost everywhere. Here \(f^\wedge\) denotes the Fourier transform of \(f\) defined on \(R\). In the present paper the Cowling-Price result is proved where \(R\) is replaced by the special unitary group \(G=SU(1,1)\) of anti-symmetric \(2\times 2\) complex matrices with determinant equal to 1. The proof makes use of the Iwasawa decomposition of \(G=KAN\), a description of the irreducible unitary representations of \(G\), and an inversion formula for rapidly decreasing \((m,n)\)-spherical functions on \(G\) whose \((m,n)\)-spherical transforms are integrable.
scientific article; zbMATH DE number 1741723