Introduction

Our subject is the set of real, integrable solutions of the inequality

where f f (x) denotes the convolution f f (x) = R d f (x -y)f (y) dy. By Young's inequality [6,Theorem 4.2], for all 1 p 2 and all f ∈ L p (R d ), f f is well defined as an element of L p/(2-p) (R d ). Thus, one may consider the inequality (1) in L p (R d ) for all 1 p 2, but the case p = 1 is special: the solution set of ( 1) is restricted in a number of surprising ways. Integrating both sides of (1), one sees immediately that R d f (x) dx 1. We prove that, in fact, all integrable solutions satisfy R d f (x) dx 1 2 , and this upper bound is sharp.

Perhaps even more surprising, we prove that all integrable solutions of (1) are nonnegative. This is not true for the solutions in L p (R d ), 1 < p 2. For f ∈ L p (R d ),

which is not integrable but belongs to L p (R) for all p > 1. By the Fourier inversion theorem f = g. Taking products, one gets examples in any dimension.

To construct a family of solutions to (1), fix a, t > 0, and define g a,t (k) = ae -2π |k|t .

By [9, Theorem 1.14],

which is satisfied for all a 1/2. Since R d f a,t (x) dx = a, this provides a class of solutions of (1) that are nonnegative and satisfy

all of which have fairly slow decay at infinity, so that in every case, Our results show that this class of examples of integrable solutions of ( 1) is surprisingly typical of all integrable solutions: every real integrable solution f of (1) is positive and satisfies (3), and if there is equality in (3), f also satisfies (4). The positivity of all real solutions of (1) in L 1 (R d ) may be considered surprising since it is false in L p (R d ) for all p > 1, as example (2) shows. We also show that when strict inequality holds in (3) for a solution f of (1), it is possible for f to have a rather fast decay; we construct examples such that R d e ε|x| f (x) dx < ∞ for some ε > 0. The conjecture that integrable solutions of (1) are necessarily positive was motivated by recent works [3,4] on a partial differential equation involving a quadratic nonlinearity of f f type, and the result proved here is the key to the proof of positivity for the solutions of this partial differential equation; see [3]. Autoconvolutions f f have been studied extensively; see [7] and the work quoted there. However, the questions investigated by these authors are quite different from those considered here.

Theorems and proofs

for all x. Then, R d f (x) dx 1 2 , and f is given by the series

which converges in L 1 (R d ) and where the c n 0 are the Taylor coefficients in the

In particular, f is positive. Moreover, if u 0 is any integrable function with

, then the sum on the right in ( 6) defines an integrable function f that satisfies (5), and

and the 1st inequality is strict for k = 0. Hence, for k = 0, √ 1 -4 u(k) = 0. By the Riemann-Lebesgue theorem, f (k) and u(k) are both continuous and vanish at infinity, and hence, we must have that

for all sufficiently large k, and in any case

But by continuity and the fact that √ 1 -4 u(k) = 0 for any k = 0, the sign cannot switch. Hence, ( 10)

which proves (3). The fact that c n as specified in (7) satisfies c n ∼ n -3/2 is a simple application of Stirling's formula, and it shows that the power series for √ 1z converges absolutely and uniformly everywhere on the closed unit disc.

Inverting the Fourier transform yields (6), and since

The final statement follows from the fact that if f is defined in terms of u in this manner, then (10) is valid, and then (8) and ( 5) are satisfied. Suppose temporarily that in addition, |x| 2 w(x) is integrable. Let σ 2 be the variance of w that is,

By the central limit theorem, since ϕ is bounded and continuous,

where γ (x) is a centered Gaussian probability density with variance σ 2 .

This shows that there is a δ > 0 such that for all sufficiently large n, R d |x| n w(x) dx √ nδ, and then since

To remove the hypothesis that w has finite variance, note that if w is a probability density with zero mean and infinite variance, 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, lim n→∞ |x| R n w(n 1/2 x)n d/2 dx = 0 (12) so that the limit in (11) has the value 1. The proof then proceeds as above. The fact that (12) is valid is a consequence of Lemma 6 below, which is closely based on the proof of [2, Corollary 1].

Proof. We may suppose that f is not identically 0. Let t := 4 R d u(x) dx 1. Then, t > 0. Define w := t -1 4u; w is a probability density and

By hypothesis, w has a zero mean and variance

By Hölder's inequality, for all 0 < p < 2, R d |x| p n w(x) dx (nσ 2 ) p/2 . It follows that for 0 < p < 1,

again using the fact that c n ∼ n -3/2 . Remark 4. In the subcritical case R d f (x) dx < 1 2 , the hypothesis that R d xu(x) dx = 0 is superf luous, and one can conclude more. In this case, the quantity t in (14) satisfies 0 < t < 1, and if we let m denote the mean of w,

Finally, the final statement of Theorem 1 shows that critical case functions f satisfying the hypotheses of Theorem 2 are readily constructed.

Theorem 2 implies that when f = 1 2 , f cannot decay faster than |x| -(d+1) . However, integrable solutions f of (1) such that R d f (x) dx < 1 2 can decay more rapidly, as indicated in the previous remark. In fact, they may even have finite exponential moments, as we now show. Consider a nonnegative, integrable function u, which integrates to r < 1 4 and satisfies

for some λ > 0. The Laplace transform of u is u(p) := e -px u(x) dx, which is analytic for |p| < λ, and u(0) < 1 4 . Therefore, there exists 0 < λ 0 λ such that, for all |p| λ 0 ,

is an integrable solution of (1). For |p| λ 0 , it has a well-defined Laplace transform f (p) given by cosh(dsx j ). Thus, for |s| < δ := λ 0 /d, R d cosh( dsx j )f (x) dx < ∞ for each j, and hence |s| < δ,

However, there are no integrable solutions of (1) that have compact support:

we have seen that all solutions of (1) are nonnegative, and if A is the support of a nonnegative integrable function, the Minkowski sum A + A is the support of f f . Remark 5. One might also consider the inequality f f f in L 1 (R d ), but it is simple to construct solutions that have both signs. Consider any radial Gaussian probability density g. Then, g g(x) g(x) for all sufficiently large |x|, and taking f := ag for a sufficiently large, we obtain f < f 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 f everywhere.

We close with a lemma validating (12) that is closely based on a construction in [2].

Lemma 6. Let w be a mean zero, infinite variance probability density on R d . Then, for all R > 0, (12) is valid.

Proof. Let X 1 , . . . , X n be n independent samples from the density w, and let B R denote the centered ball of radius R. The quantity in (12) is p n,R := P(n -1/2 n j=1 X j ∈ B R ). Let X 1 , . . . , X n be another n independent samples from the density w, independent of the 1st n. Then, also p n,R := P(-n -1/2 n j=1 X j ∈ B R ). By the independence and the triangle inequality,

The random variable X 1 -X 1 has a zero mean, an infinite variance, and an even density.

Therefore, without loss of generality, we may assume that w(x) = w(-x) for all x.

Pick ε > 0, and choose a large value σ 0 such that (2πσ It is then easy to find mutually independent random variables X, Y, and α such that X takes values in A, has zero mean and variance σ 2 and α is a Bernoulli variable with success probability A w(x) dx, and finally, such that αX + (1 -α)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 W n := n -1/2 n j=1 α j X j + n -1/2 n j=1 (1 -α 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 αs and the Ys, and we define n = n j=1 α j . These conditional expectations have the form E 1 B R +y n j=1 n -1/2 α j X j for some translate B R + y of B R , the ball of radius R. The sum n -1/2 n j=1 α j X j is actually the sum of n i.i.d. random variables with zero mean and variance σ 2 /n. The probability that n is significantly less than 3 4 n is negligible for large n; by classical estimates associated with the law of large numbers, for all n large enough, the probability that n < n/2 is no more than ε/3. Now, let Z be a Gaussian random variable with mean zero and variance σ 2 n/n, which is at least σ 2 0 when n n/2. Then, by the multivariate version [8] of the Berry-Esseen theorem [1,5], a version of the central limit theorem with rate information, there is a constant K d depending only on d such that

Since A is bounded, E|X 1 | 3 < ∞, and hence for all sufficiently large n, when n n/2,

Since this is uniform in y, we finally obtain P(W n ∈ B R ) ≤ ε for all sufficiently large n.

Since ε > 0 is arbitrary, (12) is proved.