On the convolution inequality f>f*f
Eric A. Carlen, Ian Jauslin, Elliott H. Lieb, Michael Loss
2020
Abstract
We consider the inequality \(f \geqslant f\star f\) for real functions in \(L^1(\mathbb R^d)\) where \(f\star f\) denotes the convolution of \(f\) with itself. We show that all such functions \(f\) are nonnegative, which is not the case for the same inequality in \(L^p\) for any \(1 < p \leqslant 2\), for which the convolution is defined. We also show that all solutions in \(L^1(\mathbb R^d)\) satisfy \(\int_{\mathbb R^d}f(x){\rm d}x \leqslant \textstyle\frac12\). Moreover, if \(\int_{\mathbb R^d}f(x){\rm d}x = \textstyle\frac12\), then \(f\) must decay fairly slowly: \(\int_{\mathbb R^d}x f(x){\rm d}x = \infty\), and this is sharp since for all \(r< 1\), there are solutions with \(\int_{\mathbb R^d}f(x){\rm d}x = \textstyle\frac12\) and \(\int_{\mathbb R^d}x^r f(x){\rm d}x <\infty\). However, if \(\int_{\mathbb R^d}f(x){\rm d}x = : a < \textstyle\frac12\), the decay at infinity can be much more rapid: we show that for all \(a<\textstyle\frac12\), there are solutions such that for some \(\epsilon>0\), \(\int_{\mathbb R^d}e^{\epsilonx}f(x){\rm d}x < \infty\).
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 arXiv preprint: arXiv:2002.04184.
 This article was peer reviewed for: International Mathematics Research Notices, Volume 2021, Issue 24, pages 1860418612, 2021.
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[TAMU20]: A simple equation to study interacting Bose gasses
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[HaRM20]: A simplified approach to interacting Bose gases
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[To20]: A simplified approach to interacting Bose gases
University of Toronto, Ontario, Canada, Mar 06 2020
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[Pr20]: A simplified approach to interacting Bose gases
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[VATech20]: A simplified approach to interacting Bose gases
VirginiaTech, Blacksburg, Virginia, USA, Feb 14 2020
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