On the convolution inequality $f \geq f\star f$

Abstract

We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space 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 integrable solutions $f$ satisfy $\int f(x){\rm d}x \leqslant \tfrac12$. Moreover, if $\int f(x){\rm d}x = \tfrac12$, then $f$ must decay fairly slowly: $\int |x| f(x){\rm d}x = \infty$, and this is sharp since for all $r< 1$, there are solutions with $\int f(x){\rm d}x = \tfrac12$ and $\int |x|^r f(x){\rm d}x <\infty$. However, if $\int f(x){\rm d}x = : a < \tfrac12$, the decay at infinity can be much more rapid: we show that for all $a<\tfrac12$, there are solutions such that for some $\epsilon>0$, $\int e^{\epsilon|x|}f(x){\rm d}x < \infty$.