Ian Jauslin

## On the convolution inequality f>f*f

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 non-negative, 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 $\epsilon>0$, $\int_{\mathbb R^d}e^{\epsilon|x|}f(x){\rm d}x < \infty$.

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LaTeX source:
• tarball: 20cjll-0.0.tar.gz
• git repository: 20cjll-git (the git repository contains detailed information about the changes in the paper as well as the source code for all previous versions).

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• [CJL19]: Analysis of a simple equation for the ground state energy of the Bose gas
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### Talks

This work has been presented at the following conferences:

• [TAMU20]: A simple equation to study interacting Bose gasses
Texas A&M, College Station, Texas, USA, May 15 2020
pdf, source

• [HaRM20]: A simplified approach to interacting Bose gases
Harvard University, Cambridge, Massachusetts, USA, Apr 01 2020
pdf, source

• [To20]: A simplified approach to interacting Bose gases
University of Toronto, Ontario, Canada, Mar 06 2020
pdf, source

• [Pr20]: A simplified approach to interacting Bose gases
Princeton University, New Jersey, USA, Mar 03 2020
video, pdf, source

• [VATech20]: A simplified approach to interacting Bose gases
VirginiaTech, Blacksburg, Virginia, USA, Feb 14 2020
pdf, source

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