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Uncertainty principle and \(L^{p}\)–\(L^{q}\)-sufficient pairs on noncompact real symmetric spaces. (English) Zbl 1026.43009

Summary: We consider a real semisimple Lie group \(G\) with finite center and a maximal compact subgroup \(K\) of \(G\). Let \(G=K\exp (\overline {a_+})K\) be a Cartan decomposition of \(G\). For \(x\in G\) denote by \(\|x\|\) the norm of the \({\mathfrak a}_+\)-component of \(x\) in the Cartan decomposition of \(G\). Let \(a>0\), \(b>0\) and \(1\leq p,q\leq\infty\). In this note we give necessary and sufficient conditions on \(a,b\) such that for all \(K\)-bi-invariant measurable functions \(f\) on \(G\), if \(e^{a\|x\|^2}f\in L^p(G)\) and \(e^{b\|\lambda \|^2} {\mathcal F}(f)\in L^q({\mathfrak a}^*_+)\) then \(f=0\) almost everywhere.

MSC:

43A85 Harmonic analysis on homogeneous spaces
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