Zero Sets for Spaces of Analytic Functions 3

arXiv:1705.03914v3 [math.CV] 29 Jun 2018 okn eso uy3 2018 3, July – version Working 626.AZ upre yaSafr rdaeFellowship. Graduate Stanford a by supported A.Z. 1612363. inr[ Wiener n.Is.Fuir Grenoble Fourier, Inst. Ann. oewiebt uhr eevstn irsf eerh R Research, Microsoft Founda visiting were Science authors National both while the done by supported partially R.L. (*) Fo sets. classification: zero Math. such in interested Keywords: been have physicists and maticians EOST O PCSO NLTCFUNCTIONS ANALYTIC OF SPACES FOR SETS ZERO eo fGusa nltcfntoswr rgnlysuidb ae a Paley by studied originally were functions analytic Gaussian of Zeros nebe ezeo ordsepcsd ocin analytiqu fonctions de espaces des zéros pour de Ensembles eas ieasmlrrsl nteuino w o oe such more) (or two of union the on result similar a give also We ega pcsadalwn st teghnarsl fZu( Zhu of result ( a strengthen spaces. Shapiro to Bargmann–Fock space of us on allowing Bargmann–Fock conjecture and on spaces another Bergman (1993) establishing Zhu thereby a of sets, and question spaces Bergman a on resolve (1979) to Shapiro us of conjecture a lishes ega pc rBrmn–oksaehstepoet hta that where property vanishes space the that has in function space non-zero Bargmann–Fock no or space Bergman ’xsepsd ocinnnnledn et saeqis’a qui espace propriété cette la dans q non-nulle p.s. F fonction donné a de Bargmann–Fock pas de n’existe ou Bergman de R ltqegaussienne alytique ltcfunction alytic Abstract. sae eBrmne ospretn ernocru résul Bargmann–Fock. un de renforcer espaces de les permettant sur nous (1993) Zhu et (197 u Bergman Shapiro de de aussi en d conjecture espaces tels donne autre plus) question On une (ou Bargmann–Fock. résoudre une ainsi zéros, deux de montrant réunion de de la espaces permet sur les résultat similaire nous sur (1993) et Zhu Bergman de espaces 17 ega,Brmn,Fc,Gusa,random. Gaussian, Fock, Bargmann, Bergman, esum ´ ’nue eideoteuecnetr eSaio(99 s (1979) Shapiro de conjecture démontre une Ceci s’annule. ,Kc[ Kac ], e. ´ yRselLOSadAe HI(*) ZHAI Alex and LYONS Russell by nmnr u osdscniin abe,uefnto an- fonction une faibles, conditions des sous que montre On 02,3B0 01,60G15. 30C15, 30B20, 30H20, 8 eso htudrml odtos asinan- Gaussian a conditions, mild under that show We , 9 F ,adRc [ Rice and ], htas osntbln oagvnweighted given a to belong not does a.s. that F .Introduction 1. u ’patetpsps au saepondéré espace à un p.s. pas n’appartient qui 19 , 20 .Snete,mn oemathe- more many then, Since ]. F edmond. os hsestab- This does. inudrgatDMS- grant under tion ato hswr was work this of Part ebe de sembles )srles sur 9) 99 on 1979) nl oùnnule a de tat rles ur 1993) llows zero u’il .s. n s. e some r es nd 2 RUSSELL LYONS AND ALEX ZHAI

of the history, see [26] and [7]. Those sources also give surveys of certain aspects of zero sets of Gaussian analytic functions as random objects. The topic of the present paper, however, is not mainly zero sets of Gaussian analytic functions as random objects, but as tools to understand zero sets in standard spaces of analytic functions. In particular, we consider the (weighted) Bergman spaces in the unit disk and the (weighted) Bargmann– Fock spaces in the entire plane, for which we give a unified treatment. In Subsection 1.1, we give a brief history of what is known for zero sets of functions in these spaces, focused on results relevant to ours. More can be found in Chapter 4 of [4] and Chapter 4 of [3], which are devoted to zero sets of Bergman spaces, and Chapter 5 of [29], which is devoted to zero sets of Bargmann–Fock spaces. Let µ be a finite measure on (0, ), not identically 0. Write r := ∞ µ inf r ; µ(r, )=0 (0, ], and assume that µ r = 0. For p (0, ), { ∞ } ∈ ∞ { µ} ∈ ∞ write Ap(µ) for the set of analytic functions f defined for z < r that sat- | | µ isfy rµ 1 f(re2πiθ) p dθ dµ(r) < . | | ∞ Z0 Z0 When rµ = 1, these spaces are referred to as weighted Bergman spaces, whereas when rµ = , they are called weighted Bargmann–Fock spaces. ∞p Clearly all spaces A (µ) when rµ is finite are isomorphic to weighted Bergman spaces. Denote the unit disk by D := z ; z < 1 . The un- p { | | } weighted Bergman spaces A (D) correspond to dµ(r)=2r1[0,1](r) dr. The 2 α most-studied weights are dµ(r) = 2(1 r ) 1[0,1](r) dr (α> 1), in which − p − case the corresponding Bergman spaces are denoted Aα(D). By contrast, the most-studied Bargmann–Fock spaces are defined differently, with µ de- 2 pending on p, namely, dµ(r) = pαr e−pαr /2 dr (α > 0), in which case the p corresponding Bargmann–Fock spaces are denoted Bα(C). An older defini- p tion of Bα(C)wasused by[28], where µ did not depend on p; in our notation, p p q this was the space B2α/p(C). By [29, Theorem 2.10], Bα(C) ( Bα(C) for 0

Theorem 1.1. — Let µ be a ﬁnite measure on (0, ) with µ rµ =0. ∞ 1/n { } Let p (0, ). Suppose that a > 0 satisfy lim sup an < and ∈ ∞ n n→∞ ∞

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 3

r ∞ a2 r2n / Lp/2(µ). Let F (z) := ∞ a ζ zn for z < r , where 7→ n=0 n ∈ n=0 n n | | µ ζ are independent complex Gaussian random variables. Then a.s. the only n P P analytic function f Ap(µ) with Z(f) Z(F ) is f 0. ∈ ⊇ ≡ Note that if r ∞ a2 r2n / Lp/2(µ) for all p>p , then by consider- 7→ n=0 n ∈ 0 ing a countable set of p>p , we may conclude that a.s. for all p>p , the P 0 0 only analytic function f Ap(µ) with Z(f) Z(F ) is f 0. ∈ ⊇ ≡ The following corollary, in the special case where µ1 = µ2, was known for Bergman spaces [5]. The corollary follows from Theorem 1.1 and (1.7). Corollary 1.2. — Let R (0, ]. Let µ (i =1, 2) be ﬁnite measures ∈ ∞ i with r = R. Let p (0, ) (i = 1, 2). Suppose that there exist a > 0 µi i ∈ ∞ n 1/n ∞ 2 2n p1/2 that satisfy lim sup an < and r a r L (µ ) n→∞ ∞ 7→ n=0 n ∈ 1 \ Lp2/2(µ ). Then there is a function f Ap1 (µ ) such that the only g 2 ∈ P1 ∈ Ap2 (µ ) with Z(g) Z(f) is g 0. 2 ⊇ ≡ We will actually prove a quantitative version of Theorem 1.1. Write A(µ) for the set of functions that are analytic in z ; z < r . For s 6 r and { | | µ} µ f A(µ), write Z (f) for the multiset of z with f(z)=0 and 0 < z

f p := f p . k kA (µ) k kA (µ,rµ) Given a sequence a ; n > 0 , write a(r) for the sequence a rn ; n > 0 h n i h n i and a(r) for its ℓ2-norm. k k2 −1 1/n Theorem 1.3. — Let a > 0 satisfy R := lim sup an < and n n→∞ ∞ a = 0. Let F (z) := ∞ a ζ zn for z < R, where ζ are independent 0 6 n=0 n n | | n complex Gaussian random variables. Then for all ﬁnite measures µ with P r = R and µ R =0, all p (0, ), and all s (0, R], µ { } ∈ ∞ ∈ f(0) (1.1) E max | | ; 0 f A(µ),Z(f) Zs(F ) f Ap(µ,s) 6≡ ∈ ⊃ h nk k oi √πa 6 0 1/p . s a(r) p dµ(r) 0 k k2 Proof of Theorem 1.1 from Theorem 1.3. ConsiderR 0 f A(µ) with 6≡ ∈ Z(F ) Z(f); we will show that f p = . ⊂ k kA (µ) ∞ Without loss of generality, we may shift the indices of the an so that a = 0, since this does not aﬀect the condition a(r) Lp(µ), and it does 0 6 k k2 6∈ not change ZR(F ). Thus, Theorem 1.3 applies.

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 4 RUSSELL LYONS AND ALEX ZHAI

If f(0) = 0, then the result follows directly from (1.1) by taking s = R. 6 Otherwise, we may reduce to this case: Let m denote the order of vanishing of f at 0, and let g(z) := f(z)/zm, so that g A(µ) and g(0) = 0. We then ∈ 6 have Z(g) Z (F ), from which we conclude that g p = . This ⊃ R k kA (µ) ∞ clearly implies that also f p = , as desired. k kA (µ) ∞ We also establish the following theorem, which relates to unions of zero sets in the special case b 1 upon observing that Z(F N 1) = N−1 Z(F ≡− − k=0 − e2πik/N ). S Theorem 1.4. — Let µ be a finite measure on (0, ) with µ rµ =0. ∞ 1/n { } Let p (0, ). Suppose that a > 0 satisfy lim sup an < and ∈ ∞ n n→∞ ∞ r ∞ a2 r2n / Lp/2(µ). Let F (z) := ∞ a ζ zn for z < r , where 7→ n=0 n ∈ n=0 n n | | µ ζ are independent complex Gaussian random variables. Let b A(µ) and n P P ∈ N be a positive integer. Then a.s. the only analytic function f Ap/N (µ) ∈ with Z(f) Z(F N + b) is f 0. ⊇ ≡ This establishes the full conjecture of Shapiro [25] and enlarges the set of b to which it applies. Since Z(F 1) are a.s. simple (see [18, Lemma 28]) ± and Z(F 1) are disjoint, we obtain the following corollary. ± Corollary 1.5. — Let R (0, ]. Let µ (i = 1, 2) be finite mea- ∈ ∞ i sures with rµi = R and µi R = 0. Let pi (0, ) (i = 1, 2). Sup- { } ∈ ∞ 1/n pose that there exist a > 0 that satisfy lim sup an < and n n→∞ ∞ r ∞ a2 r2n Lp1/2(µ ) Lp2/2(µ ). Then there are functions f ,f 7→ n=0 n ∈ 1 \ 2 1 2 ∈ Ap1 (µ ) such that Z(f ) Z(f ) = ∅ and the only g Ap2/2(µ ) with P1 1 ∩ 2 ∈ 2 Z(g) Z(f ) Z(f ) is g 0. ⊇ 1 ∪ 2 ≡ Again, we prove a quantitative version of Theorem 1.4: −1 1/n Theorem 1.6. — Let a > 0 satisfy R := lim sup an < . n n→∞ ∞ Let F (z) := ∞ a ζ zn for z < R, where ζ are independent complex n=0 n n | | n Gaussian random variables. Then for all finite measures µ with r = R and P µ µ R = 0, all p (0, ), all b A(µ), all positive integers N, and all { } ∈ ∞ ∈ s (0, R], ∈ 1/N f(0) N E max | | ; 0 f A(µ),Z(f) Zs(F + b) f 1/N 6≡ ∈ ⊃ h nk kAp(µ,s) oi c 6 1/p , s a(r) p dµ(r) 0 k k2 where R 4N−1 1/4N 2N 1 4N c := a4N (2N)!+4 b(0) 2a2N N!+ b(0) 4 Γ − . 0 | | 0 | | 4N 1 −

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 5

Therefore, if a(r) / Lp(µ), then a.s. every f A(µ) with Z(f) Z(F N + k k2 ∈ ∈ ⊃ b) satisﬁes f p = . k kA (µ) ∞

Of course, what allows Gaussian series to have these properties is that such series have many zeros. A quantitative form of this property is what lies behind our results. Recall that by the arithmetic mean-geometric mean inequality (or Jensen’s inequality) and Jensen’s formula, every f A(µ) ∈ with f(0) = 0 satisﬁes 6

In general, this inequality can be very far from an equality; for two simple −1 examples, consider f(z) := (1 z)−1 and p > 2 or f(z) := e(1−z) and − all p. What we will show, in contrast, is that for f = F , a.s. finiteness of the right-hand side of (1.2) implies a.s. finiteness of the left-hand side and even finiteness of the expectation of the left-hand side. This is reminiscent of Fernique’s theorem (Appendix A), but the functional on the right-hand side does not satisfy the hypotheses of Fernique’s theorem. Moreover, Fer- nique’s theorem gives finiteness of a moment defined in terms of the original functional, whereas here, the Ap(µ)-norm is, as we just illustrated, not in any way a function of the right-hand side.

−1 1/n Theorem 1.7. — Let a > 0 satisfy R := lim sup an < and n n→∞ ∞ a = 0. Let F (z) := ∞ a ζ zn for z < R, where ζ are independent 0 6 n=0 n n | | n complex Gaussian random variables. Then for all ﬁnite measures µ with P r = R and µ R =0 and all p (0, ), the following are equivalent: µ { } ∈ ∞ R 1 F 2πiθ p (i) 0 exp 0 log (re ) dθ dµ(r) < a.s.; p | | ∞ (ii) E F p < ; R k kAR (µ) ∞ R 1 (iii) E exp log F (re2πiθ) p dθ dµ(r) < ; 0 0 | | ∞ R 1 (iv) Rexp Rlog F (re2πiθ) p dθ dµ(r) < with positive probability. 0 0 | | ∞ R R

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 6 RUSSELL LYONS AND ALEX ZHAI

Moreover, for all s (0, R], ∈ F (0) (1.3) E | | 1/p " s exp 1 log F (re2πiθ) p dθ dµ(r) # 0 0 | | R R √π Γ(1 + p/2)1/p a 6 0 . p 1/p E F p k kA (µ,s) The equivalence shown here may be surprising; indeed, in discussing his conjecture, Shapiro [25] wrote that the arithmetic mean-geometric mean inequality “seems to give away too much.”

1.1. History of Zero Sets

Given a collection A of analytic functions, say that Z is an A-zero set if there is some function in A whose zero set equals Z. There is no geometric characterization known for a set of points in D to be an Ap(D)-zero set, but there are necessary conditions known that are not far from known suﬃcient conditions. It is also known that no condition depending solely on the moduli of the points can be both necessary and suﬃcient. For further discussion, let z be a countable multiset in D and write

ϕz(r) := 1 z . z∈z, − | | |Xz|6r The situation for zeros of Bergman functions contrasts strongly with that for the Hardy spaces, 1 Hp(D) := f A0(D) ; sup f(re2πiθ) p dθ < , { ∈ | | ∞} r<1 Z0 where for all p (0, ], the Blaschke condition ∈ ∞ ϕz(1) < ∞ is necessary and suﬃcient to be an Hp(D)-zero set. For every p (0, ), ∈ ∞ the Blaschke condition is suﬃcient to be an Ap(D)-zero set (since Hp(D) ⊂ Ap(D)), while the condition 1 z − | | < 1+ǫ −1 ∞ z log 1 z z∈X\{0} − | | is known to be necessary for every ǫ> 0 but not for ǫ = 0 [5]. On the other hand, if a subset of z lies on a line (or in a Stolz angle), then the Blaschke

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 7

condition for that subset is also necessary for z to be an Ap(D)-zero set [23]. Combining the preceding results, we deduce that the moduli alone do not determine whether a point set is an Ap(D)-zero set. A set W is called a set of uniqueness for A(µ) if the only f A(µ) ∈ with Z(f) W is f 0. Horowitz [5] showed that for 0 1 ordered so that z is ∈ h k i | k| increasing and f(0) = 0, then 6 n 1 (1.4) sup n−1/q < , n zk ∞ kY=1 | | whereas for every p < q, there is some f Ap(D) with zero set z ; k > 1 ∈ h k i ordered so that z is increasing and f(0) = 0 satisfying | k| 6 n 1 sup n−1/q = . n zk ∞ kY=1 | | (Since (1.4) depends only on the moduli, it is not sufficient to be a zero set.) This distinction among the zero sets for different p was refined by Shapiro [25]: for 0 p A (D). Shapiro [25] did this by using random (Gaussian) series, as we detail soon.(2) S Later works [13, 1, 16] considered random angles for fixed moduli, culmi- nating in the following result. Theorem 1.8. — Let 0 1 D. Let θ be ∞ h n i⊂ n independent uniform [0, 1] random variables. If there exists ǫ> 0 such that 1 (1.5) epϕz(r) log(1+ǫ)(1 r)−1 dr < , − ∞ Z0 then a.s. z e2πiθn ; n > 1 is an Ap(D)-zero set. If q>p, then the condition h n i (1.5) is not sufficient for z e2πiθn ; n > 1 to be a.s. an Aq(D)-zero set. h n i The Blaschke condition shows that the union of two Hp(D)-zero sets is again an Hp(D)-zero set. Horowitz [5] also showed that although the union of two Ap(D)-zero sets is an Ap/2(D)-zero set (trivially: just multiply the

(1) In that paper, [24] is cited for the first proof of this existence. However, he seems to have misinterpreted the order of quantifiers. Instead, the novelty of [24] was to extend the allowed set of weights from those in [5]. (2) Actually, there was a gap in his proof: in the middle of page 168 where the quantity I(r) is being bounded below, going from the integral over T to Eω(n) throws away a part that may be negative, so the inequality does not follow. Thus, it seems that our proof of Corollary 1.2 is the first valid proof of [25, Theorem 1 (i) implies (iii)].

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 8 RUSSELL LYONS AND ALEX ZHAI

functions), it need not be an Aq(D)-zero set if q > p/2. This was again strengthened by Shapiro [25] to show that it need not be an A(p/2)+(D)- zero set.(3) Many of the above results were extended to weighted Bergman spaces. For example, for (p, α) (0, ) ( 1, ), Horowitz [5] studied the zero sets p ∈ ∞ × − ∞ of the spaces Aα(D), showing that they were distinct classes of sets for pairs with distinct values of (α + 1)/p, provided that α > 0. He asked whether it sufficed that the pairs (p, α) be distinct. The proviso that α > 0 was removed by Sedletski˘ı[21]. The full question was answered affirmatively by Sevast′yanov and Dolgoborodov [22]. Our Corollary 1.2 easily establishes the full result of [22] by using Theorem 1 of [14], which implies that for α> 1, q > 0, and c > 0, − k n+1 1 ∞ q ∞ 2 −1 q c rk (1 r)α dr < iff 2−n(α+1) c < . k − ∞ k ∞ 0 n=0 n Z Xk=0 X kX=2 2 In the above expressions, we take c2k = ak and c2k+1 = 0, and we make a judicious choice of ak so that the convergence behavior is different for distinct pairs (p, α) and (p′, α′). The most interesting case is when (α + 1)/p = (α′ + 1)/p′ and p 0. On the other hand, classical results show n | n| that if z satisfies z −2 < , then there is some f Bp (C) with n | n| ∞ ∈ α Z(f)= z (see [29, Theorem 5.3]). P The paper [2] considered particular stationary random point processes 2 and showed that for p = 2, the critical density for being a B1 (C)-zero 2 set is 1. Zhu [29, p. 203] gives examples showing that a Bα(C)-zero set 2 and a Bα(C)-uniqueness set can differ by just one point, and that for all p p, q (0, ) and n (2/q, ) Z, for every nontrivial Bα(C)-zero set W , ∈ ∞ ∈ ∞ ∩ q removing any n points from W yields a Bα(C)-zero set. Our results give new proofs of results of Zhu [28, 29] and answer his p question [29, pp. 202, 209], showing that the zero sets of Bα(C) depend on p for fixed α; he had shown that they differ for differing α, whether or not p is fixed [29, Theorem 5.8]. To apply Corollary 1.2 to Zhu’s question, we

(3) The same gap as noted in the previous footnote applies to this result, but is ﬁlled by the proof of our Corollary 1.5.

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 9

use the following result of Stokes [27]: for b > 0, ∞ tn (1.6) lim e−ttb =1 . t→∞ Γ(n + b) n=0 X n F ∞ n Given α and p, set an := α /Γ(n +2/p) and (z) := n=0 anζnz , where ζ are independent complex Gaussian random variables. We apply n p P (1.6) with b =2/p and t = αr2, so that for q> 0 and as r , → ∞ q ∞ 2 qαr2 qαr2 2 4 q 2q − 2 2n − αr − 2 1− re 2 a r re 2 e r p = r p . n ≍ n=0 X Then by (1.7) and Theorem 1.1, it follows that a.s. F Bq (C) and ∈ q>p α Z(F ) is a Bp (C)-uniqueness set. α T Similarly, for b > 0 and c R, we have the asymptotic ∈ ∞ tn lim e−ttb(log t)c =1 . t→∞ Γ(n + b)(log n)c n=0 X n 4/p 2 With an := α / Γ(n +2/p)(log n) , b = 2/p, c = 4/p, and t = αr , we obtain byq a similar calculation that a.s. F Bp (C) and the only f ∈ α ∈ Bq (C) with Z(f) Z(F ) is f 0. q

p 2pα/q ⊇ 1∪ 2 is f 0; taking q := 2p gives Zhu’s result. (Note that Bq/2 (C) decreases ≡ S 2pα/q in q by [29, Corollary 2.8].)

1.2. Shapiro’s Approach

F ∞ n Consider (z) := n=0 anζnz , where ζn are IID standard complex Gaussian random variables and a > 0 satisfy lim sup a1/n 6 1. Be- P n n→∞ n cause E log+ ζ < , we also have lim sup ζ 1/n = 1 a.s. by the | 0| ∞ n→∞ | n| Borel–Cantelli lemma, whence a.s. F (z) converges for all z D to an ana- ∈ lytic function. p+ q Let µ be a ﬁnite measure with rµ = 1. Write L (µ) := q>p L (µ). Shapiro [25] showed that the following are equivalent: S (1) r a(r) Lp(µ) Lp+(µ); 7→ k k2 ∈ \ (2) a.s. F Ap(µ) Ap+(µ); ∈ \

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 10 RUSSELL LYONS AND ALEX ZHAI

(3) a.s. F Ap(µ) and the only function in Ap+(µ) that vanishes every- ∈ where that F does is the 0 function. In addition, he showed that when (1) holds, a.s. F 1 Ap(µ) and the only function in A(p/2)+(µ) ± ∈ that vanishes on Z(F 2 1) is the 0 function. − He conjectured that the following strengthening holds: r a(r) / Lp(µ)= 7→ k k2 ∈ ⇒ p F a.s. the only function in A (µ) that vanishes on Z( ) is the 0 function and the only function in Ap/2(µ) that vanishes on Z(F 2 1) is the 0 function. − More generally, he conjectured Theorem 1.4 when rµ = 1 and b satisﬁes a certain restriction. The equivalence of (1) and (2) follows from the following equivalence: (1.7) r a(r) Lp(µ) a.s. F Ap(µ) . 7→ k k2 ∈ ⇐⇒ ∈ (This equivalence is valid for r = as well.) To see this, note that for each µ ∞ z D, the random variable F (z) has the same distribution as a(|z|) ζ . ∈ k k2 0 Thus, Tonelli’s theorem yields 1 p p (r) p (1.8) E F p = E ζ a dµ(r) . k kA (µ) | 0| · k k2 Z0 The forward implication of (1.7) is now immediate. The reverse implication is a consequence of (1.8) and Fernique’s theorem, which tells us that p if F a.s. belongs to A (µ), then there exist some c0,c1 > 0 such that c1 E exp c F p < . (See Appendix A for a statement and proof of { 0 k kA (µ)} ∞ a general form of Fernique’s theorem.) The usefulness of Shapiro’s approach comes partly from his implicit ob- servation that given µ and p, there exists r ∞ a2 r2n Lp/2(µ) 7→ n=0 n ∈ \ Lq/2(µ). This follows from the lemma in Section 3 of [24], where he q>p P considers analytic functions, not just real power series. For completeness, S we give a short proof and extension here.

Proposition 1.9. — Let µ be a ﬁnite measure with 0 < rµ 6 ; if rµ ∞ r = , then assume that rn dµ(r) < for every n > 0. For all µ ∞ 0 ∞ p (0, ), there exists r ∞ a2 r2n Lp/2(µ) Lq/2(µ). ∈ ∞ 7→ R n=0 n ∈ \ q>p Proof. — For each M > 0,P let q := p +1 /M and letSs = s(M) < rµ be −pq/(q−p) close enough to rµ that µ(s, rµ) 6 M . We can ﬁnd N = N(M)

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 11

large enough so that rµ 1 rµ rNp dµ(r) > rNp dµ(r) . 2 Zs Z0 We also have by the power-mean inequality that

rµ 2 rµ 2 Nq q 2 − 2 Np p r dµ(r) > µ(s, rµ) q p r dµ(r) Zs Zs rµ 2 p > M 2 rNp dµ(r) . s Z r 2/p k µ p nk p Now, let nk := N(2 ), and choose bk > 0 so that 0 bkr dµ(r) = ∞ 1/2k. Then b2r2nk has the desired property: Write k=1 k R rµ P 1/p f := f(r) p dµ(r) . k kp | | Z0 If p > 2, then ∞ ∞ 2 2nk 2 2nk bkr 6 bkr =1 , p/2 p/2 k=1 k=1 X X while if p< 2, then ∞ ∞ p/2 p/2 2 2nk 2 2nk bnr 6 bkr < . p/2 p/2 ∞ k=1 k=1 X X At the same time, for each q>p , consider any j with q>p +1/2j. Then

∞ rµ 2 p 2 2nk > 2 2nj > 2 2j nj p bkr bj r q/2 bj 2 r dµ(r) q/2 j k=1 Zs(2 ) X rµ 2 p 2 2j 1 nj p j−2/p > bj 2 r dµ(r) =2 . 2 0 Z ∞ Since this holds for all such j, it follows that b2r2nk = . k=1 k q/2 ∞ P

2. Proofs

In this section, we prove Theorem 1.3 and then indicate the additional steps needed for the more general Theorem 1.6. At the end, we prove The- orem 1.7.

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 12 RUSSELL LYONS AND ALEX ZHAI

Proof of Theorem 1.3. Note that the density of ζ0 with respect to Lebesgue 2 | | measure on R+ is r 2re−r . It suﬃces to prove the theorem for s < r , 7→ µ since the case s = rµ follows by taking limits. Suppose that 0 / Z(f) Z(F ). Note that 0 / Z(F ) a.s. Thus, for ∈ ⊇ ∈ 0 exp log f(re2πiθ) p dθ dµ(r) 0 0 | | Z s Z p p r = f(0) max p , 1 dµ(r) 0 | | z Z z∈Zs(f) Y | | s rp > p f(0) max p , 1 dµ(r) 0 | | F z Z z∈Zs( ) | | Y s 1 = f(0)/F (0) p exp log F (re2πiθ) p dθ dµ(r) . | | | | Z0 Z0 Therefore, s 1 −1/p f(0) 2πiθ p (2.1) | | 6 a0 ζ0 exp log F (re ) dθ dµ(r) . f Ap(µ,s) | | 0 0 | | k k Z Z Recall that for each r and θ, F (re2πiθ) is a Gaussian random variable (r) (r) n with the same distribution as a ζ , where an := a r . Write k k2 0 n G (θ) := F (re2πiθ)/ a(r) , r k k2 so that Gr(θ) is a standard complex Gaussian random variable for each r and θ. H¨older’s inequality and the arithmetic mean-geometric mean inequality yield

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 13

Multiplying both sides by a ζ and using (2.1), we have 0| 0| s (r) p 1 −1 f(0) a0 ζ0 a Gr(θ) dθ dµ(r) 6 | | · 0 k k2 0 | | (2.3) | | 1+1/p . f p s k kA (µ,s) R a(r) Rp dµ(r) 0 k k2 R Recall that for each r and θ, Gr(θ) and ζ0 are both standard complex Gaussians, and (Gr(θ), ζ0) is jointly Gaussian. By a version of Slepian’s lemma due to [11], we have

E ζ G (θ) −1 6 E ζ E G (θ) −1 =1 √π . | 0| · | r | | 0| · | r | · Taking expectations in (2.3) and applying the above inequality finishes the proof, except for showing that the maximum on the left-hand side of (1.1) is achieved and is measurable. To show these properties, note first that the maximum is achieved be- cause of a standard normal-families argument (compare [3, p. 120]). Next, for a finite multiset W , let p (z) := (z w) be the monic polyno- W w∈W − mial whose zeros are W (with multiplicity). For any analytic function f Q whose zeros include W , the function f/pW is analytic. Therefore,

f(0) max | | ; 0 f A(µ),Z(f) Zs(F ) f Ap(µ,s) 6≡ ∈ ⊃ nk k o f(0)p F (0) = max | Zs( ) | ; 0 f A(µ) . fpZ (F ) p 6≡ ∈ n s A (µ,s) o

Restricting to polynomials f with rational coeﬃcients, we see that this

maximum is measurable provided pZs(F ) is measurable. Now there is a measurable set (of probability 0) where lim sup ζ 1/n > 1; off of this set, | n| Zs(F ) is finite and can be determined by looking at the values of F on a fixed, countable, dense set of points, thereby proving the desired measura- bility.

Remark 2.1. — In fact, Theorem 1.1 may be deduced directly from (2.2) without using Slepian’s lemma: Simply take expectations of both sides and use the facts that E G (θ) −1 = √π and ζ < to obtain | r | | 0| ∞ s 1 −1/p √π E exp log F (re2πiθ) p dθ dµ(r) 6 . s p 1/p 0 0 | | a(r) dµ(r) hZ Z i 0 k k2 As s r , the right-hand side tends to 0, which alreadyR gives Theorem 1.1 ↑ µ via (2.1).

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 14 RUSSELL LYONS AND ALEX ZHAI

Proof of Theorem 1.6. We may assume that a = 0. Suppose that 0 / 0 6 ∈ Z(f) Z(F ). As before, we have for 0

s 1 −1/p (2.4) exp log F (re2πiθ)N + b(re2πiθ) p/N dθ dµ(r) | | Z0 Z0 s 1 −1/p (r) p H p/N = a 2 exp log r(θ) dθ dµ(r) 0 k k 0 Z Z s a(r) p exp 1 log H (θ)−1/N dθ dµ(r) 6 0 k k2 0 r 1+1/p R s aR(r) p dµ(r) 0 k k2 s a(r) pR 1 H (θ)−1/N dθ dµ(r) 6 0 k k2 0 r 1+1/p . R s aR(r) p dµ(r) 0 k k2 We have for any β that R

E H (θ)−β = E ζN + b(re2πiθ)/ a(r) N −β . r | 0 k k2 | 2 Now ζN has density ρ: z c z−2(N−1)/N e−|z| (with respect to area mea- 0 7→ | | sure λ , for some constant c) that is decreasing in z . Therefore, given any 2 | | α C, the rearrangement inequality of Hardy and Littlewood yields ∈ P ζN < r = ρ(z) dλ (z) > ρ(z) dλ (z)= P ζN α < r , | 0 | 2 2 | 0 − | Z|z|

(2.5) E H (θ)−β 6 Γ(1 βN/2) . r −

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 15

Multiply both sides of the inequality (2.4) by aN ζN + b(0) 1/N and use | 0 0 | H¨older’s inequality to bound the resulting expectation:

E aN ζN + b(0) 1/N H (θ)−1/N | 0 0 | · r 1 4N−1 N N 4 4N H −4/(4N−1) 4N 6 E a0 ζ0 + b(0) E r(θ) | | − 1/4N 4N 1 2N 1 4N 6 a4N (2N)!+4 b(0) 2a2N N!+ b(0) 4 Γ − , 0 | | 0 | | 4N 1 − where in the last inequality, we used (2.5) with β:= 4/(4 N 1). − Proof of Theorem 1.7. We established (1.3) during the proof of Theorem 1.3, where we rely on (1.8) and the fact that E ζ p = Γ(1+ p/2) for an | 0| equivalent expression on the right-hand side. That (ii) implies (iii) follows from the arithmetic mean-geometric mean inequality. That (iii) implies (i) and (i) implies (iv) are obvious. That (iv) implies (ii) follows from (1.3) with s = R.

Appendix A. Fernique’s Theorem We present here a general version of Fernique’s theorem, not only for use in deriving the background in Subsection 1.2, but also for comparison with our Theorem 1.7. Theorem A.1. — Let V be a separable topological vector space. Let φ: V [0, ] be Borel measurable, c [1, ), and c ,c (1, ) satisfy → ∞ ∈ ∞ 1 2 ∈ ∞ for all x, y V that φ( x) = φ(x), c φ(x) 6 φ(√2 x) 6 c φ(x), and ∈ − 2 1 φ(x + y) 6 c φ(x)+ φ(y) . Let X be a random variable with values in V such that if Y has the same distribution as X and is independent of X, then φ(X), φ(Y ) has the same joint distribution as φ (X Y )/√2 , φ (X + − Y )/√2 . If P φ(X) < = 1, then there are some α,β > 0 so that β ∞ E eαφ(X) < . ∞ Proof. — Suppose that φ (x y)/√2 6 τ and φ (x+y)/√2 >t. Then − 2 φ(x y) 6 c τ and φ(x + y) > c t. Also, φ(2y) = φ(√2 y) 6 c2φ(y), − 1 2 1 whence φ(x + y) 6 cφ(x y)+ cc2φ(y). Therefore φ(y) > (c t cc τ)/(cc2). − 1 2 − 1 1 Symmetry gives the same lower bound on φ(x). It follows that P φ(X) 6 τ P φ(Y ) >t (A.1) =P φ (X Y )/√2 6 τ, φ (X + Y )/√2 >t − 2 6 Pφ(X) > (c t cc τ)/(cc2) . 2 − 1 1

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX 16 RUSSELL LYONS AND ALEX ZHAI

Choose τ < so that P φ(X) 6 τ > e/(1 + e). Deﬁne recursively t := ∞ 0 (cc /c )τ > τ and t := (cc2/c )t + t ; thus, 1 2 n+1 1 2 n 0 (cc2/c )n+1 1 t = 1 2 − t t = P φ(Y ) >t 6 P φ(X) >t , n+1 n+1 e n 1+e 6 2 whence if we write yn := e P φ(X) > tn , then yn+1 yn, and so n n y 6 y2 6 e−2 . Therefore, n 0 n 2 βn 2 n −2 −(cc1/c2) P φ(X) >c3(cc1/c2) t0 6 e = e , 2 where β := log 2/ log(cc1/c2) > 0. This means that β P φ(X) >t 6 e−c4t

for some c4 > 0 and all t > t0. With α := c4/2, the conclusion may be veriﬁed via integration by parts.

BIBLIOGRAPHY

[1] G. Bomash, “A Blaschke-type product and random zero sets for Bergman spaces”, Ark. Mat. 30 (1992), no. 1, p. 45-60. [2] G. Chistyakov, Y. Lyubarskii & L. Pastur, “On completeness of random expo- nentials in the Bargmann-Fock space”, J. Math. Phys. 42 (2001), no. 8, p. 3754- 3768. [3] P. Duren & A. Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004, x+318 pages. [4] H. Hedenmalm, B. Korenblum & K. Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000, x+286 pages. [5] C. Horowitz, “Zeros of functions in the Bergman spaces”, Duke Math. J. 41 (1974), p. 693-710. [6] ——— , “Some conditions on Bergman space zero sets”, J. Anal. Math. 62 (1994), p. 323-348. [7] J. B. Hough, M. Krishnapur, Y. Peres & B. Virag´ , Zeros of Gaussian analytic functions and determinantal point processes, University Lecture Series, vol. 51, American Mathematical Society, Providence, RI, 2009, x+154 pages. [8] M. Kac, “On the average number of real roots of a random algebraic equation”, Bull. Amer. Math. Soc. 49 (1943), p. 314-320. [9] ——— , “A correction to “On the average number of real roots of a random algebraic equation””, Bull. Amer. Math. Soc. 49 (1943), p. 938. [10] M. Kac, Probability and related topics in physical sciences, Lectures in Applied Mathematics. Proceedings of the Summer Seminar, Boulder, Colo., vol. 1957, Inter- science Publishers, London-New York, 1959, With special lectures by G. E. Uhlen- beck, A. R. Hibbs, and B. van der Pol., xiii+266 pages. [11] J.-P. Kahane, “Une in´egalit´edu type de Slepian et Gordon sur les processus gaussiens”, Israel J. Math. 55 (1986), no. 1, p. 109-110. [12] E. Kostlan, “On the spectra of Gaussian matrices”, Linear Algebra Appl. 162/164 (1992), p. 385-388, Directions in matrix theory (Auburn, AL, 1990).

ANNALES DE L’INSTITUT FOURIER ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS 17

[13] E. LeBlanc, “A probabilistic zero set condition for the Bergman space”, Michigan Math. J. 37 (1990), no. 3, p. 427-438. [14] M. Mateljevic´ & M. Pavlovic´, “Lp-behavior of power series with positive coeffi- cients and Hardy spaces”, Proc. Amer. Math. Soc. 87 (1983), no. 2, p. 309-316. [15] F. Nazarov & M. Sodin, “Fluctuations in random complex zeroes: asymptotic normality revisited”, Int. Math. Res. Not. IMRN (2011), no. 24, p. 5720-5759. [16] M. Nowak & P. Waniurski, “Random zero sets for Bergman spaces”, Math. Proc. Cambridge Philos. Soc. 134 (2003), no. 2, p. 337-345. [17] R. E. A. C. Paley & N. Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Math- ematical Society, Providence, RI, 1934, x+184 pages. [18] Y. Peres & B. Virag´ , “Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process”, Acta Math. 194 (2005), no. 1, p. 1-35. [19] S. O. Rice, “Mathematical analysis of random noise”, Bell System Tech. J. 23 (1944), p. 282-332. [20] ——— , “Mathematical analysis of random noise”, Bell System Tech. J. 24 (1945), p. 46-156. p [21] A. M. Sedletski˘ı, “Zeros of analytic functions of the classes A∞”, in Current Prob- lems in Function Theory (Russian) (Teberda, 1985), Rostov. Gos. Univ., Rostov- on-Don, 1987, p. 24-29, 177. [22] E. A. Sevast′yanov & A. A. Dolgoborodov, “Zeros of functions in weighted spaces with mixed norm”, Math. Notes 94 (2013), no. 1-2, p. 266-280, Translation of Mat. Zametki 94 (2013), no. 2, 279–294. [23] H. S. Shapiro & A. L. Shields, “On the zeros of functions with finite Dirichlet integral and some related function spaces”, Math. Z. 80 (1962), p. 217-229. [24] J. H. Shapiro, “Zeros of functions in weighted Bergman spaces”, Michigan Math. J. 24 (1977), no. 2, p. 243-256. [25] ——— , “Zeros of random functions in Bergman spaces”, Ann. Inst. Fourier (Greno- ble) 29 (1979), no. 4, p. 159-171. [26] M. Sodin, “Zeroes of Gaussian analytic functions”, in European Congress of Math- ematics, Eur. Math. Soc., Zürich, 2005, p. 445-458. [27] G. G. Stokes, “Note on the determination of arbitrary constants which appear as multipliers of semi-convergent series”, Proceedings of the Cambridge Philosophical Society VI (1889), p. 362-366. [28] K. Zhu, “Zeros of functions in Fock spaces”, Complex Variables Theory Appl. 21 (1993), no. 1-2, p. 87-98. [29] ———, Analysis on Fock spaces, Graduate Texts in Mathematics, vol. 263, Springer, New York, 2012, x+344 pages.

Russell LYONS Department of Mathematics 831 E. 3rd St. Indiana University Bloomington, IN 47405-7106 (USA) [email protected] Alex ZHAI Department of Mathematics Stanford University 450 Serra Mall, Building 380 Stanford, CA 94305 (USA) [email protected]

SUBMITTED ARTICLE : ZEROS-AIF-RESUB.TEX