Subharmoni Almost Periodi Fun tions Favorov S.Yu. Rakhnin A. V.N. Karazin Kharkiv National University, 7 4 Svobody sq., Kharkiv, 61077, Ukraine. 0 email: rakhninuniver.kharkov.ua, Sergey.Ju.Favorovuniver.kharkov.ua 0 2 2000 Mathemati s Subje t Classi(cid:28) ation: 42A75, 30B50 n a J 8 2 ] V C . h t a m [ 1 v 9 0 8 1 0 7 0 / h t a m : v i X r a 1 Subharmoni almost periodi fun tions A.V. Rakhnin, S. Yu. Favorov We prove that almost periodi ity in the sense of distributions oin ides with almost periodi ity with respe t to Stepanov's metri for the lass of subharmoni fun tions in a horizontal strip. We also prove that Fourier oe(cid:30) ients of these fun tions are ontinuous fun tions in Im z. Further, if the logarithm of a subharmoni almost periodi fun tion is a subharmoni fun tion, then it is almost periodi . 2 Subharmoni almost periodi fun tions were introdu ed in [2℄ in onne tion with in- vestigation of zero distribution of holomorphi almost periodi fun tions in a strip. In this paper almost periodi ity was de(cid:28)ned in the sense of distributions, namely as al- most periodi ity of the onvolution with a test fun tion. However, subharmoni fun tions log|f(z)| f(z) , where is a holomorphi almost periodi fun tion, were onsidered mu h earlier in papers [5℄ and [6℄, where the important point was to prove almost periodi ity of su h fun tions in the sense of distributions. In [4℄ this was extended to a subharmoni uniformly almost periodi fun tion whose logarithm is a subharmoni fun tion. In this paper we prove that subharmoni almost periodi in the sense of distributions fun tions arealmostperiodi inthe lassi alsense, ifwe onsider Stepanov integralmetri instead of the uniform metri . Therefore the lasses of subharmoni almost periodi in the sense of distributions fun tions and subharmoni Stepanov almost periodi fun tions are the same. NowtheFourier-Bohr oe(cid:30) ientsofsu hfun tions anbede(cid:28)nedintheusualway. For Imz ahorizontalstripthese oe(cid:30) ientsarefun tionsdependingon . In thispaperwe prove Imz that these oe(cid:30) ients depend ontinuously on , whi h allows us to approximate any subharmoni almost periodi fun tion by exponential sums with ontinuous oe(cid:30) ients in Stepanovmetri . Thus we prove thatsubharmoni almostperiodi fun tionsareStepanov almost periodi in the sense of the de(cid:28)nition in [8℄. exp(u) u In [2℄ it was proved that , where is a subharmoni almost periodi in the sense of distributions fun tion, is also almost periodi in the sense of distributions. Moreover, log|f(z)| f(z) |f(z)| for an almost periodi fun tion , where is a holomorphi fun tion, is uniformly almost periodi . Conversely, we prove that the logarithm of a subharmoni almost periodi fun tion is an almost periodi fun tion, provided it is a subharmoni fun tion. Thus we obtain a stronger than the one in [4℄, as well as the onverse to the result in [2℄. We start with the following de(cid:28)nitionsfa(nzd)nozt=atixon+si(ysee [1, p. 51℄)R. +iK K De(cid:28)nition 1. A oRntinuous fun tion K =({0} ), de(cid:28)ned on , where is a ompa t subset of (it is allows that {tn} ⊂)R, is alled uniformly almost per{iotdn′i} (Bohr almost periodi ),fif(fzro+mtna′n)y sequen e oneR +aniK hoose a subsequen e su h that the fun tions onverge uniformly on . Equivalent de(cid:28)nition is the following: ε > 0 L(ε) > 0 L(ε) For any there exists su h that ea h interval of length ontains a τ real number with the property sup |f(z +τ)−f(z)| < ε. z∈R+iK f(z) ∈ D′(S) S De(cid:28)nition 2. A distribution of order 0 ( is an open horizontal strip) ϕ ∈ D(S) is alled almost periodi , if for any test fun tion the onvolution u(z)ϕ(z −t)dxdy Z is uniformly almost periodi on the real axis. f(z) {hnN}o⊂teRthat a ording to [6℄, for an almo{shtnp′}eriodi distribuϕt(izo)nf(z +fhrno′m)dxandyy sequen e one an hooΓse a=su{ϕbs(ezq+uetn) :et ∈ R,,ϕsu∈ hKt}hat R K onverge K uniformly on every set , where (cid:21) is a ompa t subset of D(S) . Any subharmoni fun tion is lo ally integrable, so we an onsider it as a distribution. 3 S A lass of subharmoni almost periodi fun tions in an open strip will be denoted WAP(S) by . −∞ < α < β < +∞ Furthermore, for we de(cid:28)ne S = {z ∈ C : α ≤ Imz ≤ β}, [α,β] ImS = {Imz : z ∈ S}, u v S [α,β] and for fun tions , , whi h are integrable on horizontal intervals in , we denote 1 d (u,v) := sup |u(z +t)−v(z +t)|dt. [α,β] Z z∈S[α;β] 0 f(z) DSe(cid:28)nition3. A fun tion integrable onhorizontal inter{vhals}in⊂aRn openhorizontal n strip is alled Stepanov almost periodi , if from any sequen e one an hoose {hn′} g(z) f(z +hn′) a subsequen e and a fun tion su h that the fun tions onverge to g(z) d α,β ∈ ImS [α,β] in the topology de(cid:28)ned by seminorms , . S A lass of a subharmoni Stepanov almost periodi fun tions in an open strip will StAP(S) be denoted by . Sin e su h fun tions are Stepanov almost periodi on every line y = const u ∈ StAP(S) , for there exists the mean value T 1 M(u,y) := lim u(x+iy)dx. T→∞ 2T Z −T u To ea h su h we an asso iate Fourier-Bohr series u(z) ∼ a (u,y)eiλx, λ Xλ∈R where a (u,y) := M(ue−iλx,y). λ are Fourier-Bohr oe(cid:30) ients. u(z) ≥ 0 G ⊂DCe(cid:28)nition 4. A fluong ut(izo)n is alled logarithmi subharmoni in a domain , if the fun tion is subharmoni in this domain. It is easy to see that a logarithmi subharmoni fun tion is subharmoni . We prove the following theorems: u(z) ∈ WAP(S) u(z) ∈ StAP(S) Theorem 1. if and only if . u(z) S Theorem 2. Let be a logarithmi subharmoni fun tion in a strip . Then logu(z) ∈ WAP(S) u(z) ∈ WAP(S) if and only if . u(z) S Theorem 3. Let be a subharmoni almost periodi fun tion in a strip . Then ImS its Fourier-Bohr oe(cid:30) ients are ontinuous in . From Theorem 3 and Bessel inequality for Fouriue(rz-B) ohr oe(cid:30) ie{nλts∈itRf:olalo(wus,yt)ha6≡t λ spe trumofanalmostperiodi subharmoni fun tion (i.e. theset 0} ) it is most ountable, whi h also follows from Theorem 1.12 in [6℄. 4 u(z) S Theorem 4. Subharmoni fun tion in an open horizontal strip is almost periodi if and only if there exists a sequen e (cid:28)nite exponential sums Nm P (z) = a(m)(y)eiλ(nm)x, m n (1) Xn=1 λ ∈ R a(m)(y) ∈ C(ImS) u(z) n n where , , whi h onverges to the fun tion in the topology d α,β ∈ ImS [α,β] de(cid:28)ned by seminorms , . P (z) m = 1,2,.. S m Moreover, , are subharmoni fun tions in . To prove the theorems above we use the following propositions: D′(G) Proposition 1. Convergen e of subharmoni fun tions in is equivalent to the L1 (G) onvergen e in loc (see [7℄). Proposition 2. Weak limit of subharmoni fun tions is subharmoni fun tion (see [7℄). Gµ µ B(R,0) We denote by the Green potential of a measure for the disk , i.e. |R2 −zζ| Gµ(z) := log dµ(ζ). Z R|z −ζ| B(R,z0) µ µ n Lemma 1. Let measures onverge uniformly to a measure in a neighborhood of B(R,0) µ(∂B(R,0)) = 0 t > 0 t > 0 t2+t2 < R2 the disk , and . Then for any 1 , 2 su h that 1 2 , t1 lim sup |Gµn(z)−Gµ(z)|dx = 0, n→∞ Z (2) y∈[−t2;t2] −t1 z = x+iy where . ν = µ −µ n n P r o o f. Denote . We have t1 t1 |R2 −zζ| sup |Gµn(z)−Gµ(z)|dx ≤ sup log dν (ζ) dx+ n Z Z (cid:12)Z R (cid:12) y∈[−t2;t2] y∈[−t2;t2] (cid:12) B(R,0) (cid:12) −t1 −t1 (cid:12) (cid:12) (cid:12) (cid:12) t1 + sup log|z −ζ|dν (ζ) dx n Z (cid:12)Z (cid:12) (3) y∈[−t2;t2] (cid:12) B(R,0) (cid:12) −t1 (cid:12) (cid:12) (cid:12) (cid:12) µ(∂B(R,0)) = 0 µ n The ondition implies that the restri tions of the measures to the B(R,0) µ disk onverge weakly to the restri tion of the measure on the disk, and the log(|R2 −zζ|R−1) |x| ≤ t |y| ≤ t ζ ∈ B(R,0) 1 2 fun tion is ontinuous for , , . Thus the (cid:28)rst term on the right-hand side of (3) is small. Without loss of generality, we an assume R < 1/2 z,ζ ∈ B(R,0) log|z −ζ| < 0 that , so that for we have . ε > 0 log |z−ζ| = max{log|z−ζ|,logε} ε Let bean arbitrary(cid:28)xednumber. We denote . |x| ≤ t |y| ≤ t ζ ∈ B(R,0) ε > 0 1 2 This fun tion is ontinuous for , , and for any . We have 5 t1 t1 sup log|z −ζ|dν (ζ) dx ≤ sup log |z −ζ|dν (ζ) dx+ Z (cid:12)Z n (cid:12) Z (cid:12)Z ε n (cid:12) y∈[−t2;t2] (cid:12) B(R,0) (cid:12) y∈[−t2;t2] (cid:12) B(R,0) (cid:12) −t1 (cid:12) (cid:12) −t1 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) t1 + sup |log|z −ζ|−log |z −ζ||d|ν |(ζ)dx. ε n Z Z y∈[−t2;t2] B(R,0) −t1 n The (cid:28)rst term on the right-handside of this inequalityis smallwhen issu(cid:30) iently large. Then t1 sup |log|z −ζ|−log |z −ζ||d|ν |(ζ)dx = ε n Z Z y∈[−t2;t2] B(R,0) −t1 sup (logε−log|z −ζ|)dxd|ν |(ζ) ≤ n Z Z y∈[−t2;t2] B(R,0) [−t1;t1]∩{x:|x+iy−ζ|≤ε} ε (logε−log|x|)dxd|ν |(ζ) ≤ 2ε|ν |(B(R,0)). n n Z Z B(R,0) −ε ε > 0 ν n Note that sin e C ∈ Ris arbit|rνar|y(,Ba(Rnd,0m))e<asuCres weakly onverge to zero, one an n hoose a onstant with . The lemma is proved. u (z) G ⊂ C n Lemma 2. Let be a sequen e of subharmoni fun tions in a domain u (z) 6≡ −∞ D′(G) 0 onverging to a fun tion in , and let supu (z) ≤ W(G′) < ∞ n z∈G′ G′ ⊂ G [a;b]×[α;β] ⊂ G for any subdomain . Then for any re tangle , b lim sup |u (z)−u (z)|dx = 0. n 0 n→∞ Z (4) y∈[α;β] a B(z ,R) ⊂⊂ G 0 P r o o f. For every disk we have the following representation u (z) = −Gµn(z;z )+H (z;z ;u ) n = 0,1.2.., n R 0 R 0 n µ u (z) Gµn(z;z ) where n are the Riesz measures of the fun tions n , R 0 is the Green potential µ B(z ,R) H (z;z ;u ) n 0 R 0 n of the measure in the disk , and are the best harmoni majorants u (z) µ n n ofthefun tions inthisdisk. Conditionsofthelemmaimplythat onverge weakly µ µ(∂B(z ,R)) = 0 0 0 to the measure . Without loss of generality, we an assume that , and t ,t t2 +t2 < R2 1 2 1 2 using Lemma 1 we on lude that for any , t1+x0 sup |Gµn(z;z )−Gµ0(z;z )|dx −→ 0, Z R 0 R 0 y0−t2≤y≤y0+t2 −t1+x0 6 n → ∞ H (z;z ;u ) R 0 n when . Fromthisitfollowsthatthefun tions onvergetothefun tion H (z;z ;u ) D′(B(z ,R)) R 0 0 0 in . Now using the mean value property, Harnak inequality, and obvious inequality H (z;z ;u ) ≤ W(B(z ,R)) < ∞, n = 0,1,2.., R 0 n 0 H (z;z ;u ) H (z;z ;u ) R 0 n R 0 0 we obtain uniform onvergen e of to the fun tion in the re t- [−t + x ,t + x ] × [−t + y ,t + y ] [α,β]× [a,b] 1 0 1 0 2 0 2 0 angle . Covering the re tangle by a (cid:28)nite number of su h re tangles, we prove the lemma. StAP(S) ⊂ WAP(S) P r o o fof Theorem 1. In lusion S α,β ∈iIsmoSbvious. We prove{thhe}o⊂ppRo- [α,β] j site in lusion. We onsider arbitrary substrip , and a sequen e . u(z) {h } Sin e is a subharmoni almost periodi distribution, there exists a subsequen e jk v(z) sϕu∈ hDth(Sat for) some subharmto∈niR ( learly also almost periodi ) fun tion and for any [α,β] , uniformly in , lim (u(z +h +t)−v(z +t))ϕ(z)dxdy = 0. k→∞Z jk (5) S u(z +h ) v(z) k Now we will show that the fun tions onverge to in the topology de(cid:28)ned d α,β ∈ ImS ε > 0 α,β ∈ ImS [α,β] 0 by seminorms , . Assuming the ontrary, there exist , k′ su h that for an in(cid:28)nite sequen e d (u(z +h ),v(z)) > ε , [α,β] jk′ 0 {tk′} ∈ R and therefore there exists a subsequen e su h that 1 sup Z |u(z +hjk′ +tk′)−v(z +tk′)|dx > ε0. (6) y∈[α;β] 0 Passing to a subsequen e (if ne essary), we an assume that u(z +hjk′ +tk′) → w(z) , v(z +tk′) → w1(z) â D′(S[α,β]). Lemma 2 implies 1 sup Z |u(z +hjk′ +tk′)−w(z)|dx → 0, k′ → ∞ y∈[α,β] 0 and 1 sup |v(z +tk′)−w1(z)|dx → 0, k′ → ∞, Z y∈[α,β] 0 and thus inequality (6) implies 1 sup |w(z)−w (z)|dx ≥ ε . 1 0 Z (7) y∈[α;β] 0 ϕ(z) On the other hand, using (5), for any test fun tion , Z (w(z)−w1(z))ϕ(z)dxdy = kl′i→m∞Z (u(z +hjk′ +t′k)−v(z +tk′))ϕ(z)dxdy S[α,β] S[α,β] 7 = kl′i→m∞Z (u(z +hjk′)−v(z))ϕ(z −tk′)dxdy = 0, S[α,β] w(z) = w (z) w(z) w (z) 1 1 and thus almost everywhere. Sin e and are subharmoni fun - w(z) ≡ w (z) 1 tions, then , whi h ontradi ts (7). The theorem is proved. To prove Theorem 2 we need the following lemmas. ϕ(t) [−c,c] ε > 0 Lemma 3. Let be a fun tion ontinuous in . Then for any there δ ϕ ε K exists , depending on and , su h that for two integrable on ompa t set fun tions f,g : K → [−c,c] the inequality |f(x)−g(x)|dm < δ Z K implies the inequality |ϕ(f(x))−ϕ(g(x))|dm < ε. Z (8) K τ > 0 |t −t | < τ |ϕ(t ) −ϕ(t )| < ε P r o o f. Choose su h that 1 2 implies 1 2 2m(K), and denote A = {x ∈ K : |f(x)−g(x)| < τ}, 1 A = {x ∈ K : |f(x)−g(x)| ≥ τ}. 2 m(A ) ≤ 1 |f(x)−g(x)|dm Noti e that 2 τ A2 , and therefore R |ϕ(f(x))−ϕ(g(x))|dm ≤ |ϕ(f(x))−ϕ(g(x))|dm+ |ϕ(f(x))−ϕ(g(x))|dm ≤ Z Z Z K A1 A2 m(A )ε 2sup|ϕ(t)| 1 ≤ + |f(x)−g(x)|dm. 2m(K) τ Z K δ Choosing suitable , (8) follows. The lemma is proved. u (z) n Lemma 4. Let be a sequGen⊂ eCof uniformly bounded from aubo(vze) l6≡oga0rithmi 0 subharmoni fun tions in a domain , onverging to a fun tion in the logu (z) logu (z) n 0 sense of distributions. Then the fun tions onverge to the fun tion in the sense of distributions. u (z) n P r o o f. The fun tions are logarithmi subharmoni , and in parti ular subhar- u (z) u (z) L1 (G) moni . Using Proposition 1, n onverge to 0 in loc . V > 0 Next, these fun tions are uniformly bounded from above by some onstant , 0 l (t) = logmax{ε,t} ε bounded from below by , and the fun tion is ontinuous in the [0,V] ε l (u )(z) ε n interval . Lemma 3 implies that for (cid:28)xed the fun tions onverge to the l (u )(z) L1 (G) fun tion ε 0 in loc , and thus in the sense of distributions. From Proposition 2 l (u )(z) ε ε 0 it follows that the fun tions are subharmoni for all , and their monotone limit ε → 0 logu 0 when , i.e. the fun tion , is also subharmoni . B(z ,r) ⊂⊂ G L1(B(z ,r)) 0 0 Now we onsider a disk . From the onvergen e in of the un(z) {un′(z)} sequen e itfollowsthatthesubsequen e onverges uniformlyon every (cid:28)xed 8 K ⊂ B(z ,r) logu (z) 1 0 0 ompa t set with positive Lebesgue measure. Sin e the fun tion −∞ K 1 is subharmoni and not identi ally on , sup(logu (z)) ≥ C , 0 0 z∈K1 or sup(u (z)) ≥ eC0. 0 z∈K1 n′ > n 0 Thus for all sup(un′(z)) ≥ eC0−1, z∈K1 and sup(logun′(z)) ≥ C0 −1, ∀n = 0,1.. z∈K1 u (z) G n Sin e the fun tions are uniformly bounded from above on ompa t subsets of , it {logun′(z)} D′(G) follows that the family is ompa t in . Therefore there exists a subse- quen e logun′′(z) whi h onverges in D′(G) (and also in L1loc(G)) to some subharmoni v(z) G fun tion in . K ⊂ G ε > 0 Note that for any ompa t set and for any we have the following inequality |max{logun′(z),logε}−max{v(z),logε}|dxdy ≤ |logun′(z)−v(z)|dxdy. Z Z K K max{logun′(z),logε} max{v(z),logε} Hen e the fun tions onverge to the fun tion L1 (G) ε > 0 in loc for any . l (u )(z) l (u )(z) L1 (G) On the other hand, as was shown above, ε n onverge to ε 0 in loc . Thus almost everywhere (and, sin e the fun tions are subharmoni , everywhere) max{v(z),logε} = max{logu (z),logε}. 0 (9) −∞ Sin e a set on whi h a subharmoni fun tion equals to has Lebesgue ε → 0 mes({z ∈ G : v(z) < logε}) → 0 measure zero, then implies that , mes({z ∈ G : logu (z) < logε}) → 0 v(z) = logu (z) 0 0 , and almost everywhere, and logun′(z) hen e everywhere. Thus the sequen e of the fun tions onverges to the fun tion logu (z) D′(G) L1 (G) 0 in and in loc . logu (z) ε > 0 K ∈ G If for some subsequen e of the fun tions nj , 0 and ompa t set 0 |logu (z)−logu (z)|dxdy ≥ ε , Z nj 0 0 (10) K0 u (z) then, using the above onstru tion of the sequen e nj , we have that some subsequen e {logu } logu (z) L1 (G) of the sequen e nj onverges to 0 in loc , whi h ontradi ts (10). The lemma is proved. P r o o f of Theorem 2. From Proposition 3 in [2℄ it follows that the in lusion logu ∈ WAP(S) u ∈ WAP(S) impul(iezs) t∈haWt AinP l(uSsi)on {h } ⊂ R . We are going to show the op- n posite in lusion. Let and be an arbitrary sequen e. Passing 9 u 0 to a subsequetn∈ eRif ne essary, we an assume that for some subharmoni fun tion , uniformly in , lim (u(z +h )−u (z))ϕ(z −t)dxdy = 0. n 0 n→∞Z (11) S t ∈ R To prove the theorem it is su(cid:30) ient to verify that uniformly in lim logu(z +h +t)ϕ(z)dxdy = logu (z +t)ϕ(z)dxdy. n 0 n→∞Z Z (12) S S ε > 0 t → ∞ n Assuming that this fails, for some and some sequen e , logu(z +h +t )ϕ(z)dxdy − logu (z +t )ϕ(z)dxdy ≥ ε. n n 0 n (cid:12)Z Z (cid:12) (13) (cid:12) S S (cid:12) (cid:12) (cid:12) u (z)(cid:12) u (z) ∈ WAP(cid:12) (S) 0 0 Here is a logarithmi subharmoni fun tion with . Passing to a u (z) 0 subsequen e and using almost periodi ity of the fun tion , we an assume that lim u (z +t )ϕ(z)dxdy = v(z)ϕ(z)dxdy 0 n n→∞Z Z (14) S S v(z) tfo∈r sRome subharmoni in the strip S fun tion . Sin e the limit in (11) is uniform in , (14) implies lim u(z +h +t )ϕ(z)dxdy = v(z)ϕ(z)dxdy. n n n→∞Z Z S S logv(z)ϕ(z)dxdy Now Lemma 4 implies that both integrals in (13) have the same limit , n → ∞ R when , whi h is impossible. Thus (12) holds and Theorem 2 is proved. S P r o o f of Theorem 3. Without loss of generality, we an assume that is a strip S S ⊂⊂ S u(z) 0 0 with (cid:28)nite width. Let be an arbitrary substrip, . Sin e the fun tion µ := 1 ∆u is almost periodi , its Riesz measure 2π is also almost periodi in the sense of distributions. Denote 1 K(w) = log|e−γw2 −1|, 2 π where 0 < γ < . max (y −y )2 1 2 y1,y2∈ImS K(w) Note that the kernel is a subharmoni fun tion whi h is bounded from above in S S 0 and its restri tion to satis(cid:28)es the equation ∆K(w) = 2πδ(w), (15) δ(w) where is a standard Dira measure. Denote V(z) = K(w −z)ϕ(Imw)dµ(w), Z (16) S ϕ ≥ 0 ImS ϕ(y) = 1 y ∈ ImS 0 where is a test fun tion on su h that for . 10