An \(L^p\)-\(L^q\) version of Hardy's theorem for spherical Fourier transform on semisimple Lie groups (Q1774794)

Summary: We consider a real semisimple Lie group \(G\) with finite center and \(K\) a maximal compact subgroup of \(G\). We prove an \(L^p-L^q\) version of Hardy's theorem for the spherical Fourier transform on \(G\). More precisely, let \(a\), \(b\) be positive real numbers, \(1\leq p\), \(q\leq\infty\), and \(f\) a \(K\)-bi-invariant measurable function on \(G\) such that \(h_a^{-1}f\in L^p(G)\) and \(e^{b\|\lambda\|^2}\mathcal{F}(f)\in L^q({\mathfrak{a}_+^*})\) (\(h_a\) is the heat kernel on \(G\)). We establish that if \(ab\geq1/4\) and \(p\) or \(q\) is finite, then \(f=0\) almost everywhere. If \(ab<1/4\), we prove that for all \(p\), \(q\), there are infinitely many nonzero functions \(f\) and if \(ab=1/4\) with \(p=q=\infty\), we have \(f=\text{const} h_a\).