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

PDF:

LaTeX source:
• tarball: 20cjll-1.0.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).

### 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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