×

Uncertainty principle and the \(L^p\)-\(L^q\)-version of Morgan’s theorem on some groups. (English) Zbl 1039.43011

Summary: We begin with the \(L^p\)-\(L^q\)-version of Morgan’s theorem for the real line \(\mathbb{R}\). More precisely, we prove the following result. Let \(1\leq p,q\leq +\infty\), \(\alpha > 2\), and \(\beta = \alpha/(\alpha-1)\). Then, for any measurable function \(f\) on \(\mathbb{R}\), the conditions \(e^{a| x|^\alpha} f\in L^p(\mathbb{R})\) and \(e^{b|\lambda|^\beta} \widehat f\in L^q(\mathbb{R})\) imply that \(f=0\) if and only if \((a\alpha)^{1/\alpha}(b\beta)^{1/\beta}>(\sin\frac\pi2(\beta-i))^{1/\beta}\). We extend this result to the Euclidean space \(\mathbb{R}^n\), the Heisenberg group, and the noncompact real symmetric spaces.

MSC:

43A80 Analysis on other specific Lie groups
22E25 Nilpotent and solvable Lie groups
42B30 \(H^p\)-spaces
PDFBibTeX XMLCite