## An uncertainty inequality involving $L^1$ norms

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- by Enrico Laeng and Carlo Morpurgo PDF
- Proc. Amer. Math. Soc.
**127**(1999), 3565-3572 Request permission

## 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
- MR Author ID: 295007
- Email: enrlae@mate.polimi.it
**Carlo Morpurgo**- Affiliation: Dipartimento di Matematica, Università degli Studi di Milano, 20133 Milano, Italy
- Email: morpurgo@dsdipa.mat.unimi.it
- 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
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**127**(1999), 3565-3572 - MSC (1991): Primary 26D15, 42A82
- DOI: https://doi.org/10.1090/S0002-9939-99-05022-4
- MathSciNet review: 1621969