On the Convolution Inequality f ≥ f ⋆ f

We consider the inequality $f \geqslant f\star f$ for real integrable functions on $d$ dimensional Euclidean space where $f\star denotes convolution of $f$ with itself. show that all such are non-negative, which is not case same in $L^p$ any $1 < p \leqslant 2$, defined. also solutions satisfy $\int f(x){\rm d}x \tfrac12$. Moreover, if = \tfrac12$, then must decay fairly slowly: |x| \infty$, and this sharp since $r< 1$, there \tfrac12$ |x|^r <\infty$. However, : a at infinity can be much more rapid: we $a<\tfrac12$, some $\epsilon>0$, e^{\epsilon|x|}f(x){\rm \infty$.