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An uncertainty inequality involving $L^{1}$ norms

Authors: Enrico Laeng and Carlo Morpurgo
Journal: Proc. Amer. Math. Soc. 127 (1999), 3565-3572
MSC (1991): Primary 26D15, 42A82
Published electronically: May 17, 1999
MathSciNet review: 1621969
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Abstract: We derive a sharp uncertainty inequality of the form

\begin{equation*}\|x^{2} f\|_{1}^{} \,\|\xi \;\,\hat {\!\!f}\|_{2}^{2}\ge {\frac{\Lambda _{0}}{4\pi ^{2}}}\, \|f\|_{1}^{}\,\|f\|_{2}^{2},\end{equation*}

with $\Lambda _{0}=0.428368\dots $. As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum $\lambda _{0}^{}$ of the functional $\lambda (p)=4\pi ^{2} \|x^{2} p\|_{1}^{}\|x^{2} \hat p\|_{1}^{}/(p(0)\hat p(0))$, taken over all $p\ge 0$ with $\hat p\ge 0\;$ ($p\not \equiv 0$). Precisely, we obtain that $\lambda _{0}^{}\le 2\Lambda _{0}=0.85673673\dots ,$ which improves a previous bound of T. Gneiting.

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Additional Information

Enrico Laeng
Affiliation: Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy

Carlo Morpurgo
Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milano, Italy

Received by editor(s): February 13, 1998
Published electronically: May 17, 1999
Additional Notes: The second author was partially supported by NSF grant DMS-9622891.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 1999 American Mathematical Society