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 $1 < p \leqslant2$, for which the convolution is defined. We also show that all solutions in $L^1(\mathbb R^d)$ satisfy $\int_{\mathbb R^d}f(x)dx \leqslant\textstyle\frac12$. Moreover, if
$\int_{\mathbb R^d}f(x)dx =\textstyle\frac12$, then $f$ must decay fairly slowly: $\int_{\mathbb R^d}|x| f(x)dx =\infty$, and this is sharp since for all
$r< 1$, there are solutions with $\int_{\mathbb R^d}f(x)dx =\textstyle\frac12$ and $\int_{\mathbb R^d}|x|^r f(x)dx <\infty$. However, if
$\int_{\mathbb R^d}f(x)dx = : a < \textstyle\frac12$, the decay at infinity can be much more rapid: we show that for all $a<\textstyle\frac12$, there are solutions such that for some $\epsilon>0$,
where $f\star f(x)$ denotes the convolution $f\star f(x)=\int_{\mathbb R^d} f(x-y) f(y)dy$. By Young's inequality \cite[Theorem 4.2]{LL96}, for all $1\leqslant p\leqslant2$ and all $f\in L^p(\mathbb R^d)$, $f\star f$ is well defined as an element of
$L^{p/(2-p)}(\mathbb R^d)$. Thus, one may consider this inequality in $L^p(\mathbb R^d)$ for all $1\leqslant p \leqslant2$, but the case $p=1$ is special:
the solution set of (\ref{ineq}) is restricted in a number of surprising ways. Integrating both sides of (\ref{ineq}), one sees immediately that $\int_{\mathbb R^d} f(x)dx \leqslant1$.
We prove that, in fact, all integrable solutions satisfy $\int_{\mathbb R^d} f(x)dx \leqslant\textstyle\frac12$, and this upper bound is sharp.
Perhaps even more surprising, we prove that all integrable solutions of (\ref{ineq}) are non-negative. This is {\em not true} for solutions in $L^p(\mathbb R^d)$, $1 < p \leqslant2$.
For $f\in L^p(\mathbb R^d)$, $1\leqslant p \leqslant2$, the Fourier transform $\widehat{f}(k)=\int_{\mathbb R^d}e^{-i2\pi k\cdot x} f(x)dx$ is well defined as an element of $L^{p/(p-1)}(\mathbb R^d)$. If $f$ solves the equation $f = f\star f$,
then $\widehat{f}=\widehat{f}^2$, and hence $\widehat{f}$ is the indicator function of a measurable set. By the Riemann-Lebesgue Theorem, if $f\in L^1(\mathbb R^d)$, then
$\widehat{f}$ is continuous and vanishes at infinity, and the only such indicator function is the indicator function of the empty set. Hence the only integrable solution of
$f = f\star f$ is the trivial solution $f=0$. However, for $1 < p \leqslant2$, solutions abound: take $d=1$ and define $g$ to be the indicator function of the interval $[-a,a]$. Define
which is not integrable, but which belongs to $L^p(\mathbb R)$ for all $p> 1$. By the Fourier Inversion Theorem $\widehat{f}= g$. Taking products, one gets examples in any dimension.
To construct a family of solutions to (\ref{ineq}), fix $a,t> 0$, and define $g_{a,t}(k)= a e^{-2\pi |k| t}$. By \cite[Theorem 1.14]{SW71},
which is satisfied for all $a \leqslant1/2$. Since $\int_{\mathbb R^d}f_{a,t}(x)dx =a$, this provides a class of solutions of (\ref{ineq}) that are non-negative and satisfy
Our results show that this class of examples of integrable solutions of (\ref{ineq}) is surprisingly typical of {\em all} integrable solutions: every real integrable solution
$f$ of (\ref{ineq}) is positive, satisfies (\ref{half}),
and if there is equality in (\ref{half}), $f$ also satisfies (\ref{tail}). The positivity of all real solutions of (\ref{ineq}) in $L^1(\mathbb R^d)$ may be considered surprising since it is
false in $L^p(\mathbb R^d)$ for all $p > 1$, as the example (\ref{exam}) shows. We also show that when strict inequality holds in (\ref{half}) for a solution $f$ of (\ref{ineq}), it is possible for
$f$ to have rather fast decay; we construct examples such that $\int_{\mathbb R^d}e^{\epsilon|x|}f(x)dx < \infty$ for some $\epsilon> 0$. The conjecture that integrable solutions of (\ref{ineq})
are necessarily positive was motivated by recent work \cite{CJL20,CJL20b} on a partial differential equation involving a quadratic nonlinearity of $f\star f$ type, and the result proved here is the key to
the proof of positivity for solutions of this partial differential equations; see \cite{CJL20}. Autoconvolutions $f\star f$ have been studied extensively; see \cite{MV10} and the work quoted there. However, the questions investigated by these authors are quite different from those considered here.
In particular, $f$ is positive. Moreover, if $u\geqslant0$ is any integrable function with $\int_{\mathbb R^d}u(x)dx \leqslant\textstyle\frac14$, then the sum on the right in (\ref{fun}) defines an integrable function $f$ that satisfies (\ref{uineq}), and $\int_{\mathbb R^d}f(x) dx =\frac12$ if and only if $\int_{\mathbb R^d}u(x)dx=\frac14$.
\indent\underline{Proof}: Note that $u$ is integrable. Let $a :=\int_{\mathbb R^d}f(x)dx$ and $b :=\int_{\mathbb R^d}u(x)dx \geqslant0$. Fourier transforming,
At $k=0$, $a^2- a =-b$, so that $\left(a -\textstyle\frac12\right)^2=\textstyle\frac14- b$. Thus $0\leqslant b \leqslant\textstyle\frac14$. Furthermore, since $u \geqslant0$,
and the first inequality is strict for $k\neq0$. Hence for $k\neq0$, $\sqrt{1-4\widehat u(k)}\neq0$. By the Riemann-Lebesgue Theorem, $\widehat{f}(k)$ and $\widehat{u}(k)$ are both
continuous and vanish at infinity, and hence we must have that
for all sufficiently large $k$, and in any case $\widehat f(k)=\frac12\pm\frac12\sqrt{1-4\widehat u(k)}$. But by continuity and the fact that $\sqrt{1-4\widehat u(k)}\neq0$ for any $k\neq0$, the sign cannot switch.
Hence (\ref{hatf}) is valid for all $k$, including $k=0$, again by continuity. At $k=0$, $a =\textstyle\frac12-\sqrt{1-4b}$, which proves (\ref{half}).
The fact that $c_n$ as specified in (\ref{3half}) satisfies $c_n \sim n^{-3/2}$ is a simple application of Stirling's formula, and it shows that the power series for $\sqrt{1-z}$ converges absolutely and
uniformly everywhere on the closed unit disc. Since $|4\widehat u(k)| \leqslant1$,
\sqrt{1-4 \widehat u(k)} = 1 -\sum_{n=1}^\infty c_n (4 \widehat u(k))^n}$. Inverting the Fourier transform, yields (\ref{fun}), and since $\int_{\mathbb R^d} 4^n\star^n u(x)dx \leqslant 1$,
the convergence of the sum in $L^1(\mathbb R^d)$ follows from the convergence of $\sum_{n=1}^\infty c_n$. The final statement follows from the fact that if $f$ is defined in terms of $u$ in this manner, then (\ref{hatf}) is
\indent\underline{Proof}: If $\int_{\mathbb R^d} f(x)\ dx=\textstyle\frac12$, $\int_{\mathbb R^d}4u(x)\ dx=1$, then $w(x)=4u(x)$ is a probability density, and we can write $f(x)=\frac12\sum_{n=0}^\infty\star^n w$. Aiming for a contradiction, suppose that $|x|f(x)$ is integrable. Then $|x|w(x)$ is integrable. Let $m:=\int_{\mathbb R^d}xw(x)d x$. Since first moments add under convolution, the trivial inequality $|m||x| \geqslant m\cdot x$ yields
This shows that there is a $\delta> 0$ such that for all sufficiently large $n$, $\int_{\mathbb R^d}|x| \star^n w(x)dx \geqslant\sqrt{n}\delta$, and then since $c_n\sim n^{3/2}$, $\sum_{n=1}^\infty c_n \int_{\mathbb R^d}|x| \star^n w(x)dx=\infty$.
To remove the hypothesis that $w$ has finite variance, note that if $w$ is a probability density with zero mean and infinite variance, $\star^n w(n^{1/2}x)n^{d/2}$ is ``trying'' to converge to a ``Gaussian of infinite variance''. In particular, one would expect that for all $R>0$,
\end{equation} so that the limit in (\ref{CLT}) has the value $1$. The proof then proceeds as above. The fact that (\ref{CLT2}) is valid is a consequence of Lemma~\ref{CLTL} below, which is closely based on the proof of \cite[Corollary 1]{CGR08}.
\indent\underline{Proof}: We may suppose that $f$ is not identically $0$. Let $t :=4\int_{\mathbb R^d}u(x)dx \leqslant1$. Then $t> 0$. Define $w := t^{-1}4u$; $w$ is a probability density and
In the subcritical case $\int_{\mathbb R^d}f(x)dx < \frac12$, the hypothesis that $\int_{\mathbb R^d} x u(x) dx =0$ is superfluous, and one can conclude more. In this case the quantity $t$ in (\ref{tfor}) satisfies $0 < t < 1$, and if we let $m$ denote the mean of $w$,
$\int_{\mathbb R^d} |x|^2\star^n w(x)dx =n^2|m|^2+ n\sigma^2$. For $0<t<1$, $\sum_{n=1}^\infty n^2 c_n t^n < \infty$ and we conclude that $\int_{\mathbb R^d} |x|^2 f(x)dx < \infty$. Finally, the final statement of Theorem~\ref{theo:positivity} shows that critical case functions $f$ satisfying the hypotheses of Theorem~\ref{theo:decay} are readily constructed.
Theorem \ref{theo:decay} implies that when $\int f=\frac12$, $f$ cannot decay faster than $|x|^{-(d+1)}$. However, integrable solutions $f$ of (\ref{ineq}) such that $\int_{\mathbb R^d}f(x)dx < \textstyle\frac12$
can decay more rapidly, as indicated in the previous remark. In fact, they may even have finite exponential moments, as we now show.
However, there are no integrable solutions of (\ref{ineq}) that have compact support: We have seen that all solutions of (\ref{ineq}) are non-negative, and if $A$ is the support of a non-negative integrable function, the Minkowski sum $A+A$ is the support of $f\star f$.
One might also consider the inequality $f \leqslant f \star f$ in $L^1(\mathbb R^d)$, but it is simple to construct solutions that have both signs. Consider any radial Gaussian probability density $g$,
Then $g\star g(x)\geqslant g(x)$ for all sufficiently large $|x|$, and taking $f:= ag$ for $a$ sufficiently large, we obtain $f< f\star f$ everywhere. Now on a small neighborhood of the origin, replace the value of
$f$ by $-1$. If the region is taken small enough, the new function $f$ will still satisfy $f < f\star f$ everywhere.
We close with a lemma validating (\ref{CLT2}) that is closely based on a construction in \cite{CGR08}.
\theo{Lemma}\label{CLTL}
Let $w$ be a mean zero, infinite variance probability density on $\mathbb R^d$. Then for all $R>0$, (\ref{CLT2}) is valid.
\endtheo
\indent\underline{Proof}: Let $X_1,\dots,X_n$ be $n$ independent samples from the density $w$, and let $B_R$ denote the centered ball of radius $R$. The quantity in (\ref{CLT2}) is $p_{n,R} :=\mathbb{P}(n^{-1/2}\sum_{j=1}^n X_j\in B_R)$.
Let $\widetilde X_1,\dots,\widetilde X_n$ be another $n$ independent samples from the density $w$, independent of the first $n$. Then also $p_{n,R} :=\mathbb{P}(-n^{-1/2}\sum_{j=1}^n \widetilde X_j\in B_R)$. By the independence and the triangle inequality,
The random variable $X_1-\widetilde X_1$ has zero mean and infinite variance and an even density. Therefore, without loss of generality, we may assume that $w(x)= w(-x)$ for all $x$.
Pick $\epsilon>0$, and choose a large value $\sigma_0$ such that $(2\pi\sigma_0^2)^{-d/2}R^d|B| < \epsilon/3$, where $|B|$ denotes the volume of the unit ball $B$. The point of this is that if
$G$ is a centered Gaussian random variable with variance {\em at least}$\sigma_0^2$, the probability that $G$ lies in {\em any} particular translate $B_R+y$ of the ball of radius $R$ is no more than $\epsilon/3$. Let $A\subset\mathbb R^d$ be a centered cube such that
and note that since $A$ and $w$ are even, ${\displaystyle\int_{A} x w(x)dx =0}$.
It is then easy to find mutually independent random variables $X$, $Y$ and $\alpha$ such that
$X$ takes values in $A$ and, has zero mean and variance $\sigma^2$, $\alpha$ is a Bernoulli variable with success probability $\int_{A}w(x)dx$, and finally such that $\alpha X +(1-\alpha)Y $ has the probability density $w$. Taking independent identically distributed (i.i.d.) sequences of such random variables, $w(n^{1/2}x)n^{d/2}$ is the probability density of
${\displaystyle W_n := n^{-1/2}\sum_{j=1}^n \alpha_j X_j + n^{-1/2}\sum_{j=1}^n(1-\alpha_j)Y_j}$, and we seek to estimate
the expectation of $1_{B_R}(W_n)$. We first take the conditional expectation, given the values of the $\alpha$'s and the $Y$'s, and we define $\hat{n}=\sum_{j=1}^n\alpha_j$. These conditional expectations have the form
for some translate $B_R +y$ of $B_R$, the ball of radius $R$. The sum $n^{-1/2}\sum_{j=1}^n \alpha_j X_j$ is actually the sum of $\hat{n}$ i.i.d. random variables with mean zero and variance $\sigma^2/n$. The probability that $\hat{n}$ is significantly less than $\frac34 n$ is negligible for large $n$; by classical estimates associated with the Law of Large Numbers, for all $n$ large enough, the probability that $\hat{n} < n/2$ is no more than $\epsilon/3$. Now let $Z$ be a Gaussian random variable with mean zero and variance
$\sigma^2\hat{n}/n$ which is at least $\sigma^2_0$ when $\hat{n}\geqslant n/2$. Then by the multivariate version \cite{R19} of the Berry-Esseen Theorem \cite{B41,E42}, a version of the Central Limit Theorem
with rate information, there is a constant $K_d$ depending only on $d$ such that
Since this is uniform in $y$, we finally obtain ${\mathbb P}(W_n \in B_R)\leqslant\epsilon$ for all sufficiently large $n$. Since $\epsilon>0$ is arbitrary, (\ref{CLT2}) is proved.
\qed
We close by thanking the anonymous referee for useful suggestions.
\bibitem[B41]{B41} A. Berry, {\em The Accuracy of the Gaussian Approximation to the Sum of Independent Variates}. Trans. of the A.M.S. {\bf 49} (1941),122--136.
\bibitem[CGR08]{CGR08} E.A. Carlen, E. Gabetta and E. Regazzini, {\it Probabilistic investigation on explosion of solutions of the Kac equation with infinite initial energy}, J. Appl. Prob. {\bf 45} (2008), 95--106
\bibitem[CJL20]{CJL20} E.A. Carlen, I. Jauslin and E.H. Lieb, {Analysis of a simple equation for the ground state energy of the Bose gas}, Pure and Applied Analysis, 2020, in press, arXiv preprint arXiv:1912.04987.
\bibitem[CJL20b]{CJL20b} E.A. Carlen, I. Jauslin and E.H. Lieb, {Analysis of a simple equation for the ground state of the Bose gas II: Monotonicity, Convexity and Condensate Fraction}, arXiv preprint arXiv:2010.13882.
\bibitem[E42]{E42} C.-G. Esseen, {\em A moment inequality with an application to the central limit theorem}. Skand. Aktuarietidskr. {\bf 39} (1942) 160--170.
\bibitem[MV10]{MV10} M. Matolcsi, C. Vinuesa, {\it Improved bounds on the supremum of autoconvolutions}, J. Math. Anal. Appl. {\bf 372} (2010) 439--447.
\bibitem[R19]{R19} M. Rai\v c, {\em A multivariate Berry-Esseen Theorem with explicit constants}, Bernoulli {\bf 25} (2019) 2824--2853
\bibitem[SW71]{SW71} E. Stein and G. Weiss, {\em Introduction to Fourier analysis on Euclidean spaces}, Princeton University Press, Princeton NJ, 1971.