- Email: [email protected]
On possible deterioration of smoothness under the operation of convolution
C. R. Acad. Sci. Paris, t. 333, Série I, p. 291–296, 2001 Analyse mathématique/Mathematical Analysis
On possible deterioration of smoothness under the operation of convolution Alexander IL’INSKII Department of Mathematics, Kharkov National University, 4, Svoboda Sq., 61077, Kharkov, Ukraine E-mail: [email protected] (Reçu le 18 juin 2001, accepté le 25 juin 2001)
Abstract.
Let µ be a finite non-negative Borel measure on the real line R. We give a condition which is necessary and sufficient for the existence of an entire function p satisfying the following conditions: (i) p(x) 0 for x ∈ R, (ii) p ∈ L1 (R), (iii) ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ for every non-empty interval I ⊂ R. We give also a sufficient condition for the existence of an entire function p of finite order ρ > 1 and normal type satisfying conditions (i)–(iii). 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Sur la détérioration possible de régularité sous l’opération de convolution Résumé.
Soit µ une mesure borélienne, positive sur R et finie. Nous donnons une condition nécessaire et suffisante pour l’existence de fonctions entières p satisfaisant les conditions suivantes : (i) p(x) 0 pour x ∈ R, (ii) p ∈ L1 (R), (iii) ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R. Pour chaque ρ > 1, nous donnons aussi une condition suffisante pour l’existence de fonctions entières p d’ordre ρ > 1 et d’un type normal possédant les propriétés (i)–(iii). 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée On sait qu’en général, l’opération de la convolution améliore la régularité. Mais comme D.A. Raikov [2] l’a démontré, l’opération de convolution peut détériorer de la régularité. Il a construit des fonctions p1 , p2 , qui sont des restrictions à R de fonctions entières, dont la convolution p1 ∗ p2 n’est pas analytique partout sur R. M. Uluda˘g [3,4] a démontré que la détérioration peut être plus grande que celle de l’exemple de D.A. Raikov (voir aussi [1]). Le résultat suivant est démontré dans [1]. T HÉORÈME A. – Soit t ∈ C1 [0, +∞) une fonction telle que t (r) ↑ +∞ pour r ↑ ∞. Il existe une fonction entière p possédant les propriétés suivantes : (i) 0 < lim supr→∞ M (r, p) exp(−t(r)) < ∞, où M (r, p) = max{|p(z)| : |z| = r} ; (ii) p(x) 0 pour x ∈ R ; Note présentée par Jean-Pierre K AHANE. S0764-4442(01)02058-4/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
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(iii) p ∈ L1 (R) ; (iv) ess sup{(p ∗ p)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R. En particulier, pour tout ρ > 1, il existe une fonction entière p d’ordre ρ et d’un type normal possédant les propriétés (ii)–(iv) du théorème A. Nous nous intéressons ici au problème suivant. Soit µ une mesure borélienne sur R. Quelles conditions garantissent l’existence de fonctions entières (ou entières d’ordre ρ et d’un type normal) p avec les propriétés suivantes : p(x) 0 pour x ∈ R, p ∈ L1 (R) et ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R ? Dans cette Note, nous indiquons de telles conditions. Introduisons quelques notations : M est l’ensemble de toutes les mesures de probabilités boréliennes sur R ; P est l’ensemble de toutes les fonctions non négatives sur R telles que p ∈ L1 (R) et p 1 = 1 ; Uµ est l’ensemble de toutes les fonctions p ∈ P possédant la propriété suivante : pour chaque intervalle I non vide, l’égalité ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ est valable (µ ∈ M) ; E est l’ensemble de toutes les fonctions entières ; E ρ est l’ensemble de toutes les fonctions entières d’ordre fini ρ et d’un type normal ; D(µ) := {x ∈ R : µ({x}) > 0} ; Mµ (x) := sup{µ(Iε (x))/(2ε) : ε > 0} (où Iε (x) := (x − ε, x + ε), µ ∈ M et x ∈ R) est la fonction maximale de Hardy–Littlewood de la mesure µ. Le résultat essentiel de cette Note est le théorème suivant : T HÉORÈME 1. – Soit µ ∈ M. Alors les implications (2) et (3) sont vraies. Les théorèmes suivants sont des conséquences immédiates du théorème 1. T HÉORÈME 2. – Soit µ ∈ M, alors Uµ ∩ L∞ loc (R) = ∅ si et seulement si la condition (4) est vraie. T HÉORÈME 2 . – Soit µ ∈ M, alors Uµ ∩ E = ∅ si et seulement si la condition (4) est vraie. C OROLLAIRE 1. – Soit µ ∈ M et soit D(µ) non borné, alors Uµ ∩ E = ∅. Le résultat suivant donne une condition suffisante pour l’existence des fonctions entières p ∈ Uµ ∩ E ρ . T HÉORÈME 3. – Soient µ ∈ M, ρ > 1 et supposons qu’il existe une constante B > 1 et deux suites xn → ∞ et ∆n ↓ 0 telles que lim ∆−1 n µ I∆n (xn ) = +∞ et
n→∞
−ρ/(ρ−1) = +∞, lim x2n ∆−1 n exp −BCρ ∆n
n→∞
où Cρ := (ρ − 1)ρ−ρ/(ρ−1) . Alors Uµ ∩ E ρ = ∅. Remarque. – Nous rappelons que P ∩ E ρ = ∅ pour ρ < 1. Dans le cas ρ = 1, on peut indiquer une condition suffisante pour l’existence de fonctions p entières d’ordre 1 et d’un type maximal et telles que p ∈ Uµ . C OROLLAIRE 2. – Soit µ ∈ M, ρ > 1. Supposons D(µ) non borné et qu’il existe une suite {xn } telle que xn ∈ D(µ), xn → ∞ (n → ∞) et µ({xn })(log |xn |)(ρ−1)/ρ → ∞ (n → ∞). Alors Uµ ∩ E ρ = ∅.
1. Introduction and statement of results It is well known that for a given function f ∈ L1 (R) with a bounded support the convolution p ∗ f can only be smoother than p. This fact is widely used in analysis when one wants to improve the differential properties of p without substantial changes of its other properties. In 1939 D. Raikov [2] showed that the
292
On possible deterioration of smoothness
convolution p ∗ f may deteriorate the smoothness of p when f has an unbounded support. This effect was also studied later by M. Uluda˘g [3,4] and by the author [1]. The following fact was proved in [1]. T HEOREM A. – Let t ∈ C1 [0, +∞) be any real valued function such that t (r) ↑ +∞ for r ↑ ∞. There exists an entire function p satisfying the following conditions: (i) 0 < lim supr→∞ M (r, p) exp(−t(r)) < ∞, where M (r, p) = max{|p(z)| : |z| = r}; (ii) p(x) 0 for every x ∈ R; (iii) p ∈ L1 (R); (iv) ess sup{(p ∗ p)(x) : x ∈ I} = ∞ for every interval I ⊂ R. In particular, for all ρ > 1, there exists an entire fuction p of order ρ and normal type which satisfies conditions (ii)–(iv) of Theorem A. Let µ be a finite non-negative Borel measure on the real line R. It is reasonable to ask under which conditions we may guarantee the existence of an entire function p (or an entire function p with a prescribed growth on the complex plane C) such that p(x) 0 for all x ∈ R, p ∈ L1 (R), and ess sup (p ∗ µ)(x) : x ∈ I = ∞
for all non-empty intervals I ⊂ R.
(1)
The aim of this paper is to give such conditions. Without loss of generality we may assume that measures µ and functions p satisfy conditions µ(R) = 1 and p 1 = 1. We shall use the following notation: M is the set of all non-negative finite Borel measures on R; P is the set of all non-negative functions p ∈ L1 (R) with p 1 = 1; Uµ is the set of all functions p ∈ P possessing property (1) (µ ∈ M); E is the set of all entire functions on the complex plane C; E ρ is the set of all entire functions of finite order ρ and normal type; D(µ) := {x ∈ R : µ({x}) > 0}; Mµ (x) := sup{µ(Iε (x))/(2ε) : ε > 0} where Iε (x) := (x − ε, x + ε). Mµ (x) is the maximal Hardy–Littlewood function of a measure µ. Evidently, Mµ (x) = ∞ if µ({x}) > 0, Mµ (x) m(x) almost everywhere if µ is an absolutely continuous measure with a density m(x). The last inequality is valid for all continuity points x of the density m. Our main result is the following: T HEOREM 1. – Let µ ∈ M. Then the two following implications are valid: U µ ∩ L∞ loc (R) = ∅ =⇒ lim sup Mµ (x) = ∞;
(2)
lim sup Mµ (x) = ∞ =⇒ Uµ ∩ E = ∅.
(3)
x→±∞
x→±∞
As immediate consequences of Theorem 1 we have T HEOREM 2. – If µ ∈ M, then Uµ ∩ L∞ loc (R) = ∅ if and only if lim sup Mµ (x) = ∞.
(4)
x→±∞
T HEOREM 2 . – Condition (4) is necessary and sufficient for Uµ ∩ E = ∅. We note also the following C OROLLARY 1. – Let µ ∈ M and the set D(µ) be unbounded. Then Uµ ∩ E = ∅. The following theorem gives a condition which is sufficient for the existence of a function p possessing the following properties: p is an entire function of finite order ρ > 1 and normal type, and p ∈ Uµ .
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T HEOREM 3. – Let µ ∈ M and ρ > 1. Assume that there exist a constant B > 1, a real sequence xn → ∞, and a positive sequence ∆n → 0 such that lim ∆−1 n µ I∆n (xn ) = +∞ and
n→∞
−ρ/(ρ−1) = +∞, lim x2n ∆−1 n exp −BCρ ∆n
n→∞
where Cρ := (ρ − 1)ρ−ρ/(ρ−1) . Then Uµ ∩ E ρ = ∅. Remark. – As is well known, P ∩ E ρ = ∅ if ρ < 1. On the other hand, it may be stated a condition which is sufficient for the existence of an entire function p of order 1 and maximal type such that p ∈ Uµ . The following corollary can be regarded as an analogue of Corollary 1. C OROLLARY 2. – Let µ ∈ M, ρ > 1, and the set D(µ) be unbounded. If there exists a sequence xn ∈ D(µ), xn → ∞, such that (ρ−1)/ρ µ {xn } log |xn | → ∞ (n → ∞), then Uµ ∩ E ρ = ∅. 2. Proof of Theorem 1 Necessity. – Assume that condition (4) is not satisfied. Then there exist numbers A > 1 and B > 1 such that Mµ (x) B for all |x| A. By the definition of Mµ we have µ((x − ε, x + ε)) B · 2ε for |x| A and any ε > 0. Therefore µ(E) B mes(E) for every Borel set E ⊂ R \ (−A, A), where mes(·) denotes the Lebesgue measure. Let p ∈ P ∩ L∞ loc (R) and let I ⊂ R be any interval on the real line. If we denote B := ess sup{p(x − y) : x ∈ I, |y| < A}, then B < ∞ because p ∈ L∞ loc (R). It follows that (p ∗ µ)(x) = + p(x − y)µ(dy) R\(−A,A)
(−A,A)
B · µ (−A, A) + B ·
p(x − y) dy B + B R\(−A,A)
for every x ∈ I. Thus, p ∈ / Uµ . Sufficiency. – Without loss of generality we may assume that lim supx→−∞ Mµ (x) = ∞. It follows that there exists a sequence {xk }∞ k=1 such that 0 > xk ↓ −∞ (k → ∞), 1 hk := Mµ (xk ) → +∞ (k → ∞). To avoid the infinite values of hk , we introduce h∗k
:=
hk , hk < +∞, k, hk = +∞.
Obviously, 1 h∗k < ∞ for any k, h∗k → +∞ for k → ∞, M (xk ) h∗k for any k. We fix a sequence µ∞ ∞ {an }n=1 satisfying two conditions: an > 0 (n = 1, 2, . . .) and n=1 an < ∞. It follows from the property h∗k → ∞ (k → ∞) that there exists a sequence of indexes k(n) ↑ ∞ (n → ∞) such that an h∗k(n) 1
for all n = 1, 2, . . . .
(5)
Set yn := xk(n) , gn := h∗k(n) . Let ∆n be any positive number such that µ I∆n (yn ) gn ∆n .
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(6)
On possible deterioration of smoothness
We note that ∆n 1 for all n because µ is a probability measure and gn 1. Let us denote η(z) := π −1/2 exp(−z 2 ) (z ∈ C). It is clear that η ∈ P ∩ E and there exists ε0 > 0 such that η(x) ε0 for −1 x 1. Let {sj }∞ j=1 be a countable set dense in R. For every j = 1, 2, . . . , we construct a function pj ∈ P ∩ E such that (pj ∗ µ)(sj ) = ∞.
(7)
Let us pj (z) := cj
∞
−1 an(m) ∆−1 n(m) η (z + yn(m) − sj )∆n(m) ,
(8)
m=1
∞ where c−1 j = m=1 an(m) and n(m) ↑ ∞ is a sequence of indexes depending on j. It is obvious that pj ∈ P for any choice of the sequence n(m). We shall define n(m) more exactly later. We prove now that (7) is valid for any choice of {n(m)}. Indeed, ∞ ∞
−1 cj an(m) ∆−1 (pj ∗ µ)(sj ) = n(m) η (−t + yn(m) )∆n(m) µ(dt) −∞
cj
∞
m=1
an(m) ∆−1 n(m)
m=1 ∞
cj ε 0
yn(m) +∆n(m)
yn(m) −∆n(m)
η (yn(m) − t)∆−1 n(m) µ(dt)
∞
an(m) ∆−1 µ I (y ) c ε an(m) gn(m) = ∞ ∆ j 0 n(m) n(m) n(m)
m=1
m=1
according to (5). Let us prove that pj is an entire function if the sequence n(m) tends to infinity sufficiently rapidly. First we observe that there exists a constant v0 such that 2 −1 ∆ η (x + iy)∆−1 π −1/2 exp − 3x 8
(9)
for all |x| v0 , |y| |x|/2, and 0 < ∆ 1. Indeed, |η(x + iy)| π −1/2 exp(−3x2 /4) if x, y ∈ R, |y| |x|/2. Therefore |∆−1 η((x + iy)∆−1 )| π −1/2 ∆−1 exp(−3x2 /(4∆2 )) for |y| |x|/2 and ∆ > 0. We see that (9) follows from the inequality ∆2 log ∆−1 (3/8)(2 − ∆2 )x2 . Since 0 < ∆ 1, the last inequality is a consequence of the following inequality (8/3)∆2 log ∆−1 x2 . Thus the existence of v0 is proved. It follows from the condition yn ↓ −∞ (n → ∞) and from (9) that we can take the sequence n(m) −1 η((z + y − s )∆ )| 1 for |z| = x2 + y 2 m and for all m = 1, 2, . . . . We see so that |∆−1 j n(m) n(m) n(m) from this and condition an < ∞ that the series (8) converges uniformly on each disk of C. Therefore, pj ∈ E. We now give a construction of p. Let us fix a sequence {γj }∞ j=1 such that γj > 0 (j = 1, 2, . . .), ∞ −j γ = 1, γ max{|p (z)| : |z| j} 2 for j 2 and define j j j j=1 p(x) :=
∞
γj pj (x).
j=1
Evidently, p ∈ P ∩ E. Taking into account (7), we get p ∗ µ(sj ) = ∞ for all j. Let us prove that p ∈ Uµ . The
nfunction (p ∗ µ)(x) is a pointwise limit of a non-decreasing sequence of continuous functions fn (x) := −n p(x − y) µ(dy). Therefore the set IM := {x ∈ R : (p ∗ µ)(x) > M } ∩ I
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is open for any M > 0 and any non-empty interval I. Since all points sj from I belong to IM , we obtain IM = ∅. Therefore mes (IM ) > 0. ✷ Acknowledgement. The author is grateful to I.V. Ostrovskii for several helpful discussions.
References [1] Il’inskii A., On convolution of entire probability densities, Complex Variables 36 (1998) 165–181. [2] Raikov D.A., On composition of analytic distribution functions, Doklady Akad. Nauk SSSR 23 (1939) 511–514 (in Russian). [3] Uluda˘g M., On possible deterioration of smoothness under the operation of convolution, C. R. Acad. Sci. Paris, Série I 322 (1996) 173–178. [4] Uluda˘g M., On possible deterioration of smoothness under the operation of convolution, J. Math. Anal. Appl. 227 (1998) 335–358.
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On possible deterioration of smoothness under the operation of convolution Alexander IL’INSKII Department of Mathematics, Kharkov National University, 4, Svoboda Sq., 61077, Kharkov, Ukraine E-mail: [email protected] (Reçu le 18 juin 2001, accepté le 25 juin 2001)
Abstract.
Let µ be a finite non-negative Borel measure on the real line R. We give a condition which is necessary and sufficient for the existence of an entire function p satisfying the following conditions: (i) p(x) 0 for x ∈ R, (ii) p ∈ L1 (R), (iii) ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ for every non-empty interval I ⊂ R. We give also a sufficient condition for the existence of an entire function p of finite order ρ > 1 and normal type satisfying conditions (i)–(iii). 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Sur la détérioration possible de régularité sous l’opération de convolution Résumé.
Soit µ une mesure borélienne, positive sur R et finie. Nous donnons une condition nécessaire et suffisante pour l’existence de fonctions entières p satisfaisant les conditions suivantes : (i) p(x) 0 pour x ∈ R, (ii) p ∈ L1 (R), (iii) ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R. Pour chaque ρ > 1, nous donnons aussi une condition suffisante pour l’existence de fonctions entières p d’ordre ρ > 1 et d’un type normal possédant les propriétés (i)–(iii). 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
Version française abrégée On sait qu’en général, l’opération de la convolution améliore la régularité. Mais comme D.A. Raikov [2] l’a démontré, l’opération de convolution peut détériorer de la régularité. Il a construit des fonctions p1 , p2 , qui sont des restrictions à R de fonctions entières, dont la convolution p1 ∗ p2 n’est pas analytique partout sur R. M. Uluda˘g [3,4] a démontré que la détérioration peut être plus grande que celle de l’exemple de D.A. Raikov (voir aussi [1]). Le résultat suivant est démontré dans [1]. T HÉORÈME A. – Soit t ∈ C1 [0, +∞) une fonction telle que t (r) ↑ +∞ pour r ↑ ∞. Il existe une fonction entière p possédant les propriétés suivantes : (i) 0 < lim supr→∞ M (r, p) exp(−t(r)) < ∞, où M (r, p) = max{|p(z)| : |z| = r} ; (ii) p(x) 0 pour x ∈ R ; Note présentée par Jean-Pierre K AHANE. S0764-4442(01)02058-4/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés
291
A. Il’inskii
(iii) p ∈ L1 (R) ; (iv) ess sup{(p ∗ p)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R. En particulier, pour tout ρ > 1, il existe une fonction entière p d’ordre ρ et d’un type normal possédant les propriétés (ii)–(iv) du théorème A. Nous nous intéressons ici au problème suivant. Soit µ une mesure borélienne sur R. Quelles conditions garantissent l’existence de fonctions entières (ou entières d’ordre ρ et d’un type normal) p avec les propriétés suivantes : p(x) 0 pour x ∈ R, p ∈ L1 (R) et ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ pour chaque intervalle non vide I ⊂ R ? Dans cette Note, nous indiquons de telles conditions. Introduisons quelques notations : M est l’ensemble de toutes les mesures de probabilités boréliennes sur R ; P est l’ensemble de toutes les fonctions non négatives sur R telles que p ∈ L1 (R) et p 1 = 1 ; Uµ est l’ensemble de toutes les fonctions p ∈ P possédant la propriété suivante : pour chaque intervalle I non vide, l’égalité ess sup{(p ∗ µ)(x) : x ∈ I} = ∞ est valable (µ ∈ M) ; E est l’ensemble de toutes les fonctions entières ; E ρ est l’ensemble de toutes les fonctions entières d’ordre fini ρ et d’un type normal ; D(µ) := {x ∈ R : µ({x}) > 0} ; Mµ (x) := sup{µ(Iε (x))/(2ε) : ε > 0} (où Iε (x) := (x − ε, x + ε), µ ∈ M et x ∈ R) est la fonction maximale de Hardy–Littlewood de la mesure µ. Le résultat essentiel de cette Note est le théorème suivant : T HÉORÈME 1. – Soit µ ∈ M. Alors les implications (2) et (3) sont vraies. Les théorèmes suivants sont des conséquences immédiates du théorème 1. T HÉORÈME 2. – Soit µ ∈ M, alors Uµ ∩ L∞ loc (R) = ∅ si et seulement si la condition (4) est vraie. T HÉORÈME 2 . – Soit µ ∈ M, alors Uµ ∩ E = ∅ si et seulement si la condition (4) est vraie. C OROLLAIRE 1. – Soit µ ∈ M et soit D(µ) non borné, alors Uµ ∩ E = ∅. Le résultat suivant donne une condition suffisante pour l’existence des fonctions entières p ∈ Uµ ∩ E ρ . T HÉORÈME 3. – Soient µ ∈ M, ρ > 1 et supposons qu’il existe une constante B > 1 et deux suites xn → ∞ et ∆n ↓ 0 telles que lim ∆−1 n µ I∆n (xn ) = +∞ et
n→∞
−ρ/(ρ−1) = +∞, lim x2n ∆−1 n exp −BCρ ∆n
n→∞
où Cρ := (ρ − 1)ρ−ρ/(ρ−1) . Alors Uµ ∩ E ρ = ∅. Remarque. – Nous rappelons que P ∩ E ρ = ∅ pour ρ < 1. Dans le cas ρ = 1, on peut indiquer une condition suffisante pour l’existence de fonctions p entières d’ordre 1 et d’un type maximal et telles que p ∈ Uµ . C OROLLAIRE 2. – Soit µ ∈ M, ρ > 1. Supposons D(µ) non borné et qu’il existe une suite {xn } telle que xn ∈ D(µ), xn → ∞ (n → ∞) et µ({xn })(log |xn |)(ρ−1)/ρ → ∞ (n → ∞). Alors Uµ ∩ E ρ = ∅.
1. Introduction and statement of results It is well known that for a given function f ∈ L1 (R) with a bounded support the convolution p ∗ f can only be smoother than p. This fact is widely used in analysis when one wants to improve the differential properties of p without substantial changes of its other properties. In 1939 D. Raikov [2] showed that the
292
On possible deterioration of smoothness
convolution p ∗ f may deteriorate the smoothness of p when f has an unbounded support. This effect was also studied later by M. Uluda˘g [3,4] and by the author [1]. The following fact was proved in [1]. T HEOREM A. – Let t ∈ C1 [0, +∞) be any real valued function such that t (r) ↑ +∞ for r ↑ ∞. There exists an entire function p satisfying the following conditions: (i) 0 < lim supr→∞ M (r, p) exp(−t(r)) < ∞, where M (r, p) = max{|p(z)| : |z| = r}; (ii) p(x) 0 for every x ∈ R; (iii) p ∈ L1 (R); (iv) ess sup{(p ∗ p)(x) : x ∈ I} = ∞ for every interval I ⊂ R. In particular, for all ρ > 1, there exists an entire fuction p of order ρ and normal type which satisfies conditions (ii)–(iv) of Theorem A. Let µ be a finite non-negative Borel measure on the real line R. It is reasonable to ask under which conditions we may guarantee the existence of an entire function p (or an entire function p with a prescribed growth on the complex plane C) such that p(x) 0 for all x ∈ R, p ∈ L1 (R), and ess sup (p ∗ µ)(x) : x ∈ I = ∞
for all non-empty intervals I ⊂ R.
(1)
The aim of this paper is to give such conditions. Without loss of generality we may assume that measures µ and functions p satisfy conditions µ(R) = 1 and p 1 = 1. We shall use the following notation: M is the set of all non-negative finite Borel measures on R; P is the set of all non-negative functions p ∈ L1 (R) with p 1 = 1; Uµ is the set of all functions p ∈ P possessing property (1) (µ ∈ M); E is the set of all entire functions on the complex plane C; E ρ is the set of all entire functions of finite order ρ and normal type; D(µ) := {x ∈ R : µ({x}) > 0}; Mµ (x) := sup{µ(Iε (x))/(2ε) : ε > 0} where Iε (x) := (x − ε, x + ε). Mµ (x) is the maximal Hardy–Littlewood function of a measure µ. Evidently, Mµ (x) = ∞ if µ({x}) > 0, Mµ (x) m(x) almost everywhere if µ is an absolutely continuous measure with a density m(x). The last inequality is valid for all continuity points x of the density m. Our main result is the following: T HEOREM 1. – Let µ ∈ M. Then the two following implications are valid: U µ ∩ L∞ loc (R) = ∅ =⇒ lim sup Mµ (x) = ∞;
(2)
lim sup Mµ (x) = ∞ =⇒ Uµ ∩ E = ∅.
(3)
x→±∞
x→±∞
As immediate consequences of Theorem 1 we have T HEOREM 2. – If µ ∈ M, then Uµ ∩ L∞ loc (R) = ∅ if and only if lim sup Mµ (x) = ∞.
(4)
x→±∞
T HEOREM 2 . – Condition (4) is necessary and sufficient for Uµ ∩ E = ∅. We note also the following C OROLLARY 1. – Let µ ∈ M and the set D(µ) be unbounded. Then Uµ ∩ E = ∅. The following theorem gives a condition which is sufficient for the existence of a function p possessing the following properties: p is an entire function of finite order ρ > 1 and normal type, and p ∈ Uµ .
293
A. Il’inskii
T HEOREM 3. – Let µ ∈ M and ρ > 1. Assume that there exist a constant B > 1, a real sequence xn → ∞, and a positive sequence ∆n → 0 such that lim ∆−1 n µ I∆n (xn ) = +∞ and
n→∞
−ρ/(ρ−1) = +∞, lim x2n ∆−1 n exp −BCρ ∆n
n→∞
where Cρ := (ρ − 1)ρ−ρ/(ρ−1) . Then Uµ ∩ E ρ = ∅. Remark. – As is well known, P ∩ E ρ = ∅ if ρ < 1. On the other hand, it may be stated a condition which is sufficient for the existence of an entire function p of order 1 and maximal type such that p ∈ Uµ . The following corollary can be regarded as an analogue of Corollary 1. C OROLLARY 2. – Let µ ∈ M, ρ > 1, and the set D(µ) be unbounded. If there exists a sequence xn ∈ D(µ), xn → ∞, such that (ρ−1)/ρ µ {xn } log |xn | → ∞ (n → ∞), then Uµ ∩ E ρ = ∅. 2. Proof of Theorem 1 Necessity. – Assume that condition (4) is not satisfied. Then there exist numbers A > 1 and B > 1 such that Mµ (x) B for all |x| A. By the definition of Mµ we have µ((x − ε, x + ε)) B · 2ε for |x| A and any ε > 0. Therefore µ(E) B mes(E) for every Borel set E ⊂ R \ (−A, A), where mes(·) denotes the Lebesgue measure. Let p ∈ P ∩ L∞ loc (R) and let I ⊂ R be any interval on the real line. If we denote B := ess sup{p(x − y) : x ∈ I, |y| < A}, then B < ∞ because p ∈ L∞ loc (R). It follows that (p ∗ µ)(x) = + p(x − y)µ(dy) R\(−A,A)
(−A,A)
B · µ (−A, A) + B ·
p(x − y) dy B + B R\(−A,A)
for every x ∈ I. Thus, p ∈ / Uµ . Sufficiency. – Without loss of generality we may assume that lim supx→−∞ Mµ (x) = ∞. It follows that there exists a sequence {xk }∞ k=1 such that 0 > xk ↓ −∞ (k → ∞), 1 hk := Mµ (xk ) → +∞ (k → ∞). To avoid the infinite values of hk , we introduce h∗k
:=
hk , hk < +∞, k, hk = +∞.
Obviously, 1 h∗k < ∞ for any k, h∗k → +∞ for k → ∞, M (xk ) h∗k for any k. We fix a sequence µ∞ ∞ {an }n=1 satisfying two conditions: an > 0 (n = 1, 2, . . .) and n=1 an < ∞. It follows from the property h∗k → ∞ (k → ∞) that there exists a sequence of indexes k(n) ↑ ∞ (n → ∞) such that an h∗k(n) 1
for all n = 1, 2, . . . .
(5)
Set yn := xk(n) , gn := h∗k(n) . Let ∆n be any positive number such that µ I∆n (yn ) gn ∆n .
294
(6)
On possible deterioration of smoothness
We note that ∆n 1 for all n because µ is a probability measure and gn 1. Let us denote η(z) := π −1/2 exp(−z 2 ) (z ∈ C). It is clear that η ∈ P ∩ E and there exists ε0 > 0 such that η(x) ε0 for −1 x 1. Let {sj }∞ j=1 be a countable set dense in R. For every j = 1, 2, . . . , we construct a function pj ∈ P ∩ E such that (pj ∗ µ)(sj ) = ∞.
(7)
Let us pj (z) := cj
∞
−1 an(m) ∆−1 n(m) η (z + yn(m) − sj )∆n(m) ,
(8)
m=1
∞ where c−1 j = m=1 an(m) and n(m) ↑ ∞ is a sequence of indexes depending on j. It is obvious that pj ∈ P for any choice of the sequence n(m). We shall define n(m) more exactly later. We prove now that (7) is valid for any choice of {n(m)}. Indeed, ∞ ∞
−1 cj an(m) ∆−1 (pj ∗ µ)(sj ) = n(m) η (−t + yn(m) )∆n(m) µ(dt) −∞
cj
∞
m=1
an(m) ∆−1 n(m)
m=1 ∞
cj ε 0
yn(m) +∆n(m)
yn(m) −∆n(m)
η (yn(m) − t)∆−1 n(m) µ(dt)
∞
an(m) ∆−1 µ I (y ) c ε an(m) gn(m) = ∞ ∆ j 0 n(m) n(m) n(m)
m=1
m=1
according to (5). Let us prove that pj is an entire function if the sequence n(m) tends to infinity sufficiently rapidly. First we observe that there exists a constant v0 such that 2 −1 ∆ η (x + iy)∆−1 π −1/2 exp − 3x 8
(9)
for all |x| v0 , |y| |x|/2, and 0 < ∆ 1. Indeed, |η(x + iy)| π −1/2 exp(−3x2 /4) if x, y ∈ R, |y| |x|/2. Therefore |∆−1 η((x + iy)∆−1 )| π −1/2 ∆−1 exp(−3x2 /(4∆2 )) for |y| |x|/2 and ∆ > 0. We see that (9) follows from the inequality ∆2 log ∆−1 (3/8)(2 − ∆2 )x2 . Since 0 < ∆ 1, the last inequality is a consequence of the following inequality (8/3)∆2 log ∆−1 x2 . Thus the existence of v0 is proved. It follows from the condition yn ↓ −∞ (n → ∞) and from (9) that we can take the sequence n(m) −1 η((z + y − s )∆ )| 1 for |z| = x2 + y 2 m and for all m = 1, 2, . . . . We see so that |∆−1 j n(m) n(m) n(m) from this and condition an < ∞ that the series (8) converges uniformly on each disk of C. Therefore, pj ∈ E. We now give a construction of p. Let us fix a sequence {γj }∞ j=1 such that γj > 0 (j = 1, 2, . . .), ∞ −j γ = 1, γ max{|p (z)| : |z| j} 2 for j 2 and define j j j j=1 p(x) :=
∞
γj pj (x).
j=1
Evidently, p ∈ P ∩ E. Taking into account (7), we get p ∗ µ(sj ) = ∞ for all j. Let us prove that p ∈ Uµ . The
nfunction (p ∗ µ)(x) is a pointwise limit of a non-decreasing sequence of continuous functions fn (x) := −n p(x − y) µ(dy). Therefore the set IM := {x ∈ R : (p ∗ µ)(x) > M } ∩ I
295
A. Il’inskii
is open for any M > 0 and any non-empty interval I. Since all points sj from I belong to IM , we obtain IM = ∅. Therefore mes (IM ) > 0. ✷ Acknowledgement. The author is grateful to I.V. Ostrovskii for several helpful discussions.
References [1] Il’inskii A., On convolution of entire probability densities, Complex Variables 36 (1998) 165–181. [2] Raikov D.A., On composition of analytic distribution functions, Doklady Akad. Nauk SSSR 23 (1939) 511–514 (in Russian). [3] Uluda˘g M., On possible deterioration of smoothness under the operation of convolution, C. R. Acad. Sci. Paris, Série I 322 (1996) 173–178. [4] Uluda˘g M., On possible deterioration of smoothness under the operation of convolution, J. Math. Anal. Appl. 227 (1998) 335–358.
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