On the convolution inequality $f geq fstar f$

We consider the inequality $f geqslant fstar f$ for real integrable functions on $d$ dimensional Euclidean space where $fstar f$ 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 $epsilon>0$, $int e^{epsilon|x|}f(x){rm d}x < infty$.