\(L^p\) version of Hardy's theorem on semisimple Lie groups (Q2781358)

Roughly speaking, uncertainty principles on the real line say that a function \(f\) and its Fourier transform \(\widehat f\) cannot simultaneously be sharply localized. A classical result in this context is Hardy's theorem which can be stated as follows. For \(s>0\), let \(e_s(x)= e^{sx^2}\), \(x\in\mathbb{R}\). With suitable normalization of the Fourier transform, if \(f:\mathbb{R}\to\mathbb{C}\) is a measurable function and \(a,b>0\) are such that \(ab\geq 1/4\) and \(e_{a\pi} f\in L^\infty (\mathbb{R})\) and \(e_{b\pi}\widehat f\in L^\infty (\mathbb{R})\), then \(f=0\). A generalization, due to Cowling and Price, deals with the situation that \(e_{a \pi} f\in L^p(\mathbb{R})\) and \(e_{b\pi}\widehat f\in L^q(\mathbb{R})\) for some \(1\leq p\), \(q\leq\infty\), \(\min(p,q) <\infty\). There has been considerable effort to prove analogues for various classes of type I Lie groups. In this general setting explicit knowledge of the irreducible representations and the Plancherel formula is required.NEWLINENEWLINENEWLINEAnalogues of Hardy's theorem for semisimple Lie groups with finite centre have recently been established independently by several authors. In the paper under review the authors prove a Cowling-Price type theorem for semisimple Lie groups with finite centre. Since what might be the most obvious analogue of the Cowling-Price theorem for semisimple Lie groups fails for \(p>2\), an appropriate decay condition has to be found. Apart from using results on analytic functions, the proof heavily exploits the principal series representations of the semisimple Lie group.