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

Abstract

We consider the inequality $f⩾f\star f$ 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 $1<p⩽2$, for which the convolution is defined. We also show that all integrable solutions $f$ satisfy $\int f(x)\mathrm{d}x⩽\frac{1}{2}$. Moreover, if $\int f(x)\mathrm{d}x=\frac{1}{2}$, then $f$ must decay fairly slowly: $\int |x|f(x)\mathrm{d}x=\infty$, and this is sharp since for all $r<1$, there are solutions with $\int f(x)\mathrm{d}x=\frac{1}{2}$ and $\int |x{|}^{r}f(x)\mathrm{d}x<\infty$. However, if $\int f(x)\mathrm{d}x=:a<\frac{1}{2}$, the decay at infinity can be much more rapid: we show that for all $a<\frac{1}{2}$, there are solutions such that for some $\epsilon >0$, $\int e^{\epsilon |x|}f(x)\mathrm{d}x<\infty$.

This is the final version published in International Mathematics Research Notices

References in corpus (1)

• A multivariate Berry--Esseen theorem with explicit constants

Cited by in corpus (1)

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