L’INSTITUTFOURIER DE ANNALES

A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Russell LYONS & Alex ZHAI

Zero Sets for Spaces of Analytic Functions Tome 68, n^o6 (2018), p. 2311-2328.

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ZERO SETS FOR SPACES OF ANALYTIC FUNCTIONS

by Russell LYONS & Alex ZHAI (*)

Abstract. — We show that under mild conditions, a Gaussian analytic func- tionF that a.s. does not belong to a given weighted Bergman space or Bargmann–

Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro [21] on Bergman spaces and allows us to resolve a question of Zhu [24] on Bargmann–Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby estab- lishing another conjecture of Shapiro [21] on Bergman spaces and allowing us to strengthen a result of Zhu [24] on Bargmann–Fock spaces.

Résumé. — On montre que sous des conditions faibles, une fonction analy- tique gaussienneF qui n’appartient pas p.s. à un espace pondéré de Bergman ou de Bargmann–Fock donné a p.s. la propriété qu’il n’existe pas de fonction non- nulle dans cette espace qui s’annule oùF s’annule. Ceci démontre une conjecture de Shapiro [21] sur les espaces de Bergman et nous permet de résoudre une ques- tion de Zhu [24] sur les espaces de Bargmann–Fock. On donne aussi un résultat similaire sur la réunion de deux (ou plus) tels ensembles de zéros, montrant ainsi une autre conjecture de Shapiro [21] sur les espaces de Bergman et nous permettant de renforcer un résultat de Zhu [24] sur les espaces de Bargmann–Fock.

1. Introduction

Zeros of Gaussian analytic functions were originally studied by Paley and Wiener [13], Kac [7, 8], and Rice [15, 16]. Since then, many more mathematicians and physicists have been interested in such zero sets. For some of the history, see [22] and [6]. Those sources also give surveys of cer- tain aspects of zero sets of Gaussian analytic functions as random objects.

The topic of the present paper, however, is not mainly zero sets of Gauss- ian analytic functions as random objects, but as tools to understand zero

Keywords:Bergman, Bargmann, Fock, Gaussian, random.

2010Mathematics Subject Classification:30H20, 30B20, 30C15, 60G15.

(*) R.L. partially supported by the National Science Foundation under grant DMS- 1612363. A.Z. supported by a Stanford Graduate Fellowship. Part of this work was done while both authors were visiting Microsoft Research, Redmond.

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 [25], which is devoted to zero sets of Bargmann–Fock spaces.

Let µ be a finite measure on (0,∞), not identically 0. Write rµ :=

inf{r; µ(r,∞) = 0} ∈ (0,∞], and assume that µ({r_µ}) = 0. For p ∈ (0,∞), write A^p(µ) for the set of analytic functions f defined for|z|< rµ

that satisfy

Z r_µ 0

Z 1 0

f(re^2πiθ)

p dθdµ(r)<∞.

Whenrµ = 1, these spaces are referred to as weighted Bergman spaces, whereas whenr_µ =∞, they are called weighted Bargmann–Fock spaces.

Clearly all spaces A^p(µ) when rµ is finite are isomorphic to weighted Bergman spaces. Denote the unit disk by D := {z; |z| < 1}. The un- weighted Bergman spacesA^p(D) correspond to dµ(r) = 2r1^[0,1](r) dr. The most-studied weights are dµ(r) = 2(1−r²)^α1[0,1](r) dr(α >−1), in which case the corresponding Bergman spaces are denotedA^p_α(D). By contrast, the most-studied Bargmann–Fock spaces are defined differently, withµde- pending onp, namely, dµ(r) =pαr e^−pαr2/2dr(α >0), in which case the corresponding Bargmann–Fock spaces are denotedB^p_α(C). An older defini- tion ofB_α^p(C) was used by [24], whereµdid not depend onp; in our nota- tion, this was the spaceB_2α/p^p (C). By [25, Theorem 2.10],B^p_α(C)(B_α^q(C) for 0< p < q <∞.

Astandard complex Gaussianrandom variable is one whose density with respect to Lebesgue measure onCisz7→e^−|z|2/π. We always consider the zero setZ(f) of an analytic functionf as a multiset or a sequence, where each zerowis listed with its multiplicity, which ismifz7→f(z)/(z−w)^m is analytic and does not vanish atw.

Our main result is the following.

Theorem 1.1. — Letµbe a finite measure on(0,∞)withµ({rµ}) = 0.

Let p ∈ (0,∞). Suppose that a_n > 0 satisfy lim sup_n→∞a^1/nn < ∞ and r7→P∞

n=0a²_nr²ⁿ∈/L^p/2(µ). LetF(z) :=P∞

n=0anζnzⁿ for|z|< rµ, where ζ_nare independent complex Gaussian random variables. Then a.s. the only analytic functionf ∈A^p(µ)withZ(f)⊇Z(F)isf ≡0.

Note that ifr7→P∞

n=0a²_nr²ⁿ∈/L^p/2(µ) for allp > p₀, then by consider- ing a countable set ofp > p0, we may conclude that a.s. for allp > p0, the only analytic functionf ∈A^p(µ) withZ(f)⊇Z(F) isf ≡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. — LetR∈(0,∞]. Letµ_i (i= 1,2) be finite measures withrµ_i =R. Let pi ∈(0,∞)(i= 1,2). Suppose that there exist an >0 that satisfy lim sup_n→∞a^1/nn < ∞ and r 7→ P∞

n=0a²_nr²ⁿ ∈ L^p1/2(µ₁)\ L^p2/2(µ2). Then there is a function f ∈ A^p1(µ1) such that the only g ∈ A^p2(µ2)withZ(g)⊇Z(f)isg≡0.

We will actually prove a quantitative version of Theorem 1.1. WriteA(µ) for the set of functions that are analytic in{z; |z|< rµ}. For s6rµ and f ∈A(µ), writeZs(f) for the multiset ofz withf(z) = 0 and 0<|z|< s.

Denote

kfk_Ap(µ,s):=

Z s 0

Z 1 0

f(re^2πiθ)

p dθdµ(r) ^1/p

. We also abbreviate

kfk_Ap(µ):=kfk_Ap(µ,r_µ).

Given a sequencehan; n>0i, writea^(r)for the sequence hanrⁿ; n>0i andka^(r)k₂ for its`²-norm.

Theorem 1.3. — Leta_n>0satisfyR⁻¹:= lim sup_n→∞a^1/nn <∞and a0 6= 0. Let F(z) :=P∞

n=0anζnzⁿ for |z| < R, whereζn are independent complex Gaussian random variables. Then for all finite measuresµ with rµ=R andµ({R}) = 0, allp∈(0,∞), and alls∈(0, R],

(1.1) E

"

max

( |f(0)|

kfk_Ap(µ,s)

; 06≡f ∈A(µ), Z(f)⊃Z_s(F) )#

6

√π a₀ Rs

0 ka^(r)k^p₂dµ(r)1/p. Proof of Theorem 1.1 from Theorem 1.3. — Consider 0 6≡ f ∈ A(µ) withZ(F)⊂Z(f); we will show thatkfk_Ap(µ)=∞.

Without loss of generality, we may shift the indices of the a_n so that a06= 0, since this does not affect the conditionka^(r)k26∈L^p(µ), and it does not changeZ_R(F). Thus, Theorem 1.3 applies.

If f(0)6= 0, then the result follows directly from (1.1) by takings=R.

Otherwise, we may reduce to this case: Letmdenote the order of vanishing off at 0, and letg(z) :=f(z)/z^m, so thatg∈A(µ) andg(0)6= 0. We then

have Z(g) ⊃ Z_R(F), from which we conclude that kgk_Ap(µ) = ∞. This clearly implies that alsokfk_Ap(µ)=∞, as desired.

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) = SN−1

k=0 Z(F −e^2πik/N).

Theorem 1.4. — Letµbe a finite measure on(0,∞)withµ({rµ}) = 0.

Let p ∈ (0,∞). Suppose that a_n > 0 satisfy lim sup_n→∞a^1/nn < ∞ and r7→P∞

n=0a²_nr²ⁿ∈/L^p/2(µ). LetF(z) :=P∞

n=0anζnzⁿ for|z|< rµ, where ζ_n are independent complex Gaussian random variables. Letb∈A(µ)and N be a positive integer. Then a.s. the only analytic functionf ∈A^p/N(µ) withZ(f)⊇Z(F^N +b)isf ≡0.

This establishes the full conjecture of Shapiro [21] and enlarges the set ofbto which it applies. SinceZ(F±1) are a.s. simple (see [14, Lemma 28]) andZ(F ±1) are disjoint, we obtain the following corollary.

Corollary 1.5. — Let R ∈ (0,∞]. Let µ_i (i = 1,2) be finite mea- sures with rµ_i = R and µi({R}) = 0. Let pi ∈ (0,∞) (i = 1,2). Sup- pose that there exist a_n > 0 that satisfy lim sup_n→∞a^1/nn < ∞ and r7→P∞

n=0a²_nr²ⁿ∈L^p1/2(µ1)\L^p2/2(µ2). Then there are functionsf1, f2∈ A^p1(µ₁) such that Z(f₁)∩Z(f₂) = ∅ and the only g ∈ A^p2/2(µ₂) with Z(g)⊇Z(f1)∪Z(f2)isg≡0.

Again, we prove a quantitative version of Theorem 1.4:

Theorem 1.6. — Let an > 0 satisfy R⁻¹ := lim sup_n→∞a^1/nn < ∞.

LetF(z) :=P∞

n=0anζnzⁿ for|z|< R, whereζn are independent complex Gaussian random variables. Then for all finite measuresµwithr_µ=Rand µ({R}) = 0, all p∈(0,∞), allb ∈ A(µ), all positive integersN, and all s∈(0, R],

E



max







|f(0)|^1/N kfk^1/N_Ap(µ,s)

; 06≡f ∈A(µ), Z(f)⊃Zs(F^N +b)











6 c

Rs

0 ka^(r)k^p₂dµ(r)^1/p, where

c:= a^4N₀ (2N)! + 4|b(0)|²a^2N₀ N! +|b(0)|^41/4N Γ

2N−1 4N−1

^4N−1_4N . Therefore, if ka^(r)k₂ ∈/ L^p(µ), then a.s. every f ∈ A(µ) with Z(f) ⊃ Z(F^N +b)satisfieskfk_Ap(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, everyf ∈ A(µ) withf(0)6= 0 satisfies

(1.2)

kfk^p_Ap(µ)= Z r_µ

0

Z 1 0

f(re^2πiθ)

p dθdµ(r)

>

Z r_µ 0

exp Z 1

0

log

f(re^2πiθ)

p dθdµ(r)

= Z r_µ

0

|f(0)|^p Y

z∈Z(f)

max r^p

|z|^p,1

dµ(r).

In general, this inequality can be very far from an equality; for two simple examples, considerf(z) := (1−z)⁻¹ and p >2 or f(z) := e^(1−z)−1 and allp. 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, theA^p(µ)-norm is, as we just illustrated, not in any way a function of the right-hand side.

Theorem 1.7. — Leta_n>0satisfyR⁻¹:= lim sup_n→∞a^1/nn <∞and a0 6= 0. Let F(z) :=P∞

n=0anζnzⁿ for |z| < R, whereζn are independent complex Gaussian random variables. Then for all finite measuresµ with r_µ=R andµ({R}) = 0 and allp∈(0,∞), the following are equivalent:

(1) RR 0 expR1

0 log|F(re^2πiθ)|^pdθdµ(r)<∞a.s.;

(2) Eh

kFk^p_Ap(µ)

i

<∞;

(3) Eh RR

0 expR1

0 log|F(re^2πiθ)|^pdθdµ(r)i

<∞;

(4) RR 0 expR1

0 log|F(re^2πiθ)|^pdθdµ(r)<∞with positive probability.

Moreover, for alls∈(0, R], (1.3)

E







|F(0)|

Rs 0 expR1

0 log|F(re^2πiθ)|^pdθdµ(r)^1/p





6

√πΓ(1 +p/2)^1/pa0

Eh

kFk^p_Ap(µ,s)

i^1/p .

The equivalence shown here may be surprising; indeed, in discussing his conjecture, Shapiro [21] wrote that the arithmetic mean-geometric mean inequality “seems to give away too much.”

1.1. History of Zero Sets

Given a collectionAof analytic functions, say thatZ is anA-zero setif there is some function inAwhose zero set equalsZ. There is no geometric characterization known for a set of points inD to be an A^p(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, letzbe a countable multiset inDand write

ϕz(r) := X

z∈z,

|z|6r

(1− |z|).

The situation for zeros of Bergman functions contrasts strongly with that for the Hardy spaces,

H^p(D) :=

f ∈A⁰(D) ; sup

r<1

Z 1 0

f(re^2πiθ)

p dθ <∞

,

where for allp∈(0,∞], theBlaschke condition ϕz(1)<∞

is necessary and sufficient to be anH^p(D)-zero set. For everyp∈(0,∞), the Blaschke condition is sufficient to be anA^p(D)-zero set (sinceH^p(D)⊂ A^p(D)), while the condition

X

z∈z\{0}

(1− |z|)

log¹⁺(1− |z|)⁻¹ <∞

is known to be necessary for every >0 but not for= 0 [5]. On the other hand, if a subset ofzlies on a line (or in a Stolz angle), then the Blaschke condition for that subset is also necessary forzto be anA^p(D)-zero set [19].

Combining the preceding results, we deduce that the moduli alone do not determine whether a point set is anA^p(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 isf ≡0. Horowitz [5] showed that for 0 < p < q < ∞, there exists anA^p(D)-zero set that is anA^q(D)-uniqueness set. In fact, he

showed that iff ∈A^q(D) with zero sethzk; k>1iordered so that|zk| is increasing andf(0)6= 0, then

(1.4) sup

n

n^−1/q

n

Y

k=1

1

|z_k| <∞,

whereas for everyp < q, there is somef ∈A^p(D) with zero sethz_k; k>1i ordered so that|zk|is increasing andf(0)6= 0 satisfying

sup

n

n^−1/q

n

Y

k=1

1

|zk| =∞.

(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 [21]: for 0 < p < ∞, there exists f ∈ A^p(D) whose zeros are not the zeros of any function inA^p+(D), whereA^p+(D) :=S

q>pA^q(D).⁽¹⁾ Shapiro [21] did this by using random (Gaussian) series, as we detail soon.⁽²⁾ Later works [1, 10, 12] considered random angles for fixed moduli, cul- minating in the following result.

Theorem 1.8. — Let0< p <∞andz=hz_n; n>1i ⊂D. Letθ_n be independent uniform[0,1]random variables. If there exists >0such that (1.5)

Z 1 0

e^pϕz(r)log⁽¹⁺⁾(1−r)⁻¹dr <∞,

then a.s.hzne^2πiθn; n>1iis anA^p(D)-zero set. Ifq > p, then the condi- tion(1.5) is not sufficient forhzne^2πiθn; n> 1i to be a.s. an A^q(D)-zero set.

The Blaschke condition shows that the union of two H^p(D)-zero sets is again anH^p(D)-zero set. Horowitz [5] also showed that although the union of twoA^p(D)-zero sets is anA^p/2(D)-zero set (trivially: just multiply the functions), it need not be an A^q(D)-zero set if q > p/2. This was again strengthened by Shapiro [21] to show that it need not be an A^(p/2)+(D)- zero set.⁽³⁾

(1)In that paper, [20] is cited for the first proof of this existence. However, he seems to have misinterpreted the order of quantifiers. Instead, the novelty of [20] 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 overTto E^ω(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 [21, Theorem 1 (i) implies (iii)].

(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.

Many of the above results were extended to weighted Bergman spaces.

For example, for (p, α)∈(0,∞)×(−1,∞), Horowitz [5] studied the zero sets of the spacesA^p_α(D), showing that they were distinct classes of sets for pairs with distinct values of (α+ 1)/p, provided thatα>0. He asked whether it sufficed that the pairs (p, α) be distinct. The proviso that α > 0 was removed by Sedletski˘i [17]. The full question was answered affirmatively by Sevast⁰yanov and Dolgoborodov [18]. Our Corollary 1.2 easily establishes the full result of [18] by using Theorem 1 of [11], which implies that for α >−1, q >0, andck >0,

Z 1 0

∞

X

k=0

c_kr^k

!^q

(1−r)^αdr <∞ ⇐⇒

∞

X

n=0

2^−n(α+1)





2ⁿ⁺¹−1

X

k=2ⁿ

c_k





q

<∞.

In the above expressions, we takec2k = a²_k and c2k+1 = 0, and we make a judicious choice of a_k so that the convergence behavior is different for distinct pairs (p, α) and (p⁰, α⁰). The most interesting case is when (α+ 1)/p= (α⁰+ 1)/p⁰ and p < p⁰, which can be handled by takingq =p/2, a²₂n = 2^2n(α+1)/p·n^−2/p, andak= 0 whenk is not a power of 2.

Very little is known about the zero sets of functions in the Bargmann–

Fock spaces, even for p = 2. Zhu [24] showed that if f ∈ B^p_α(C) with f(0) 6= 0 and we write Z(f) = z as a sequence in increasing order of modulus, then infn|zn|/√

n >0. On the other hand, classical results show that if z satisfies P

n|zn|⁻² < ∞, then there is some f ∈ B_α^p(C) with Z(f) =z(see [25, Theorem 5.3]).

The paper [2] considered particular stationary random point processes and showed that for p = 2, the critical density for being a B₁²(C)-zero set is 1. Zhu [25, p. 203] gives examples showing that a B_α²(C)-zero set and a B_α²(C)-uniqueness set can differ by just one point, and that for all p, q∈(0,∞) andn∈(2/q,∞)∩Z, for every nontrivialB_α^p(C)-zero setW, removing anynpoints fromW yields aB^q_α(C)-zero set.

Our results give new proofs of results of Zhu [24, 25] and answer his question [25, p. 202 and 209], showing that the zero sets ofB_α^p(C) depend onpfor fixed α; he had shown that they differ for differing α, whether or notpis fixed [25, Theorem 5.8]. To apply Corollary 1.2 to Zhu’s question, we use the following result of Stokes [23]: forb>0,

(1.6) lim

t→∞e^−tt^b

∞

X

n=0

tⁿ

Γ(n+b) = 1.

Given α and p, set a_n := p

αⁿ/Γ(n+ 2/p) and F(z) := P∞

n=0a_nζ_nzⁿ, where ζn are independent complex Gaussian random variables. We ap- ply (1.6) withb= 2/pandt=αr², so that forq >0 and asr→ ∞,

re^−qαr

2 2

∞

X

n=0

a²_nr²ⁿ

!^q₂

re^−qαr

2 2

e^αr2r^−4pq₂

=r^1−2qp. Then by (1.7) and Theorem 1.1, it follows that a.s.F ∈T

q>pB_α^q(C) and Z(F) is a B_α^p(C)-uniqueness set.

Similarly, for b>0 andc∈R, we have the asymptotic

t→∞lim e^−tt^b(logt)^c

∞

X

n=0

tⁿ

Γ(n+b)(logn)^c = 1.

Withan :=

q

αⁿ/ Γ(n+ 2/p)(logn)^4/p

, b= 2/p, c= 4/p, andt=αr², we obtain by a similar calculation that a.s.F ∈ B_α^p(C) and the only f ∈ S

q<pB_α^q(C) withZ(f)⊇Z(F) isf ≡0.

We also strengthen Zhu’s result ([24] or [25, Theorem 5.4]) that there is a union of two disjointB_α^p(C)-zero sets that is aB^p_α(C)-uniqueness set.

Indeed, by Theorem 1.4 and Proposition 1.9, we can find disjointB_α^p(C)- zero setsZ1,Z2such that the onlyf ∈S

q>pB^q/2_2pα/q(C) withZ(f)⊇Z1∪Z2

isf ≡0; takingq:= 2pgives Zhu’s result. (Note thatB_2pα/q^q/2 (C) decreases inqby [25, Corollary 2.8].)

1.2. Shapiro’s Approach

Consider F(z) := P∞

n=0anζnzⁿ, where ζn are IID standard complex Gaussian random variables andan >0 satisfy lim sup_n→∞a^1/nn 61. Be- causeE

log⁺|ζ0|

<∞, we also have lim sup_n→∞|ζn|^1/n = 1 a.s. by the Borel–Cantelli lemma, whence a.s.F(z) converges for allz∈Dto an ana- lytic function.

Let µ be a finite measure with r_µ = 1. Write L^p+(µ) := S

q>pL^q(µ).

Shapiro [21] showed that the following are equivalent:

(1) r7→ ka^(r)k₂

∈L^p(µ)\L^p+(µ);

(2) a.s.F ∈A^p(µ)\A^p+(µ);

(3) a.s. F ∈A^p(µ) and the only function in A^p+(µ) that vanishes ev- erywhere thatF does is the 0 function.

In addition, he showed that when (1) holds,

a.s.F±1∈A^p(µ)and the only function inA^(p/2)+(µ)that vanishes onZ(F²−1)is the 0 function.

He conjectured that the following strengthening holds:

r7→ ka^(r)k₂

∈/L^p(µ) =⇒

a.s. the only function inA^p(µ)that vanishes onZ(F) is the 0 function and the only function inA^p/2(µ) that vanishes onZ(F²−1)is the 0 function.

More generally, he conjectured Theorem 1.4 whenrµ= 1 andb satisfies a certain restriction.

The equivalence of (1) and (2) follows from the following equivalence:

(1.7)

r7→ ka^(r)k₂

∈L^p(µ) ⇐⇒ a.s.F ∈A^p(µ).

(This equivalence is valid forr_µ=∞as well.) To see this, note that for each z∈D, the random variable F(z) has the same distribution aska^(|z|)k₂ζ0. Thus, Tonelli’s theorem yields

(1.8) Eh

kFk^p_Ap(µ)

i

=E[|ζ0|^p]· Z 1

0

ka^(r)k^p₂dµ(r).

The forward implication of (1.7) is now immediate. The reverse implica- tion is a consequence of (1.8) and Fernique’s theorem, which tells us that if F a.s. belongs to A^p(µ), then there exist some c0, c1 > 0 such that E

exp{c0kFk^c_A¹p(µ)}

<∞. (See Appendix A for a statement and proof of a general form of Fernique’s theorem.)

The usefulness of Shapiro’s approach comes partly from his implicit ob- servation that given µand p, there exists r7→P∞

n=0a²_nr²ⁿ

∈ L^p/2(µ)\ S

q>pL^q/2(µ). This follows from the lemma in Section 3 of [20], where he considers analytic functions, not just real power series. For completeness, we give a short proof and extension here.

Proposition 1.9. — Let µ be a finite measure with 0 < rµ 6 ∞; if r_µ = ∞, then assume that Rr_µ

0 rⁿdµ(r) < ∞ for every n > 0. For all p∈(0,∞), there exists r7→P∞

n=0a²_nr²ⁿ

∈L^p/2(µ)\S

q>pL^q/2(µ).

Proof. — For eachM >0, letq:=p+ 1/M and let s=s(M)< rµ be close enough tor_µ that µ(s, r_µ) 6M^{−pq/(q−p)}. We can find N =N(M) large enough so that

Z r_µ s

r^{N p}dµ(r)> 1 2

Z r_µ 0

r^{N p}dµ(r).

We also have by the power-mean inequality that Z rµ

s

r^{N q}dµ(r) ²_q

>µ(s, r_µ)^2q−p2 Z rµ

s

r^{N p}dµ(r) ²_p

>M² Z r_µ

s

r^{N p}dµ(r) ²_p

. Now, letnk :=N(2^k), and choosebk>0 so that Rrµ

0 b^p_kr^nkpdµ(r)^2/p

= 1/2^k. ThenP∞

k=1b²_kr^2nk has the desired property: Write kfk_p:=

Z rµ

0

|f(r)|^pdµ(r) ^1/p

. Ifp>2, then

∞

X

k=1

b²_kr^2nk p/2

6

∞

X

k=1

b²_kr^2nk

_p/2= 1, while ifp <2, then

∞

X

k=1

b²_nr^2nk

p/2

p/2

6

∞

X

k=1

b²_kr^2nk

p/2 p/2<∞.

At the same time, for eachq > p, consider anyj withq > p+ 1/2^j. Then

∞

X

k=1

b²_kr^2nk q/2

>

b²_jr^2nj

_q/2>b²_j2^2j Z r_µ

s(2^j)

r^njpdµ(r)

!²_p

>b²_j2^2j 1

2 Z r_µ

0

r^njpdµ(r) _p²

= 2^j−2/p. Since this holds for all suchj, it follows that

P∞

k=1b²_kr^2nk

_q/2=∞.

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.

Proof of Theorem 1.3. — Note that the density of |ζ₀| with respect to Lebesgue measure on R⁺ is r 7→ 2re^−r2. It suffices to prove the theorem fors < rµ, since the cases=rµ follows by taking limits.

Suppose that 0 ∈/ Z(f) ⊇ Z(F). Note that 0 ∈/ Z(F) a.s. Thus, for 0< s < rµ, we have a.s. by the arithmetic mean-geometric mean inequality and Jensen’s formula that

kfk^p_Ap(µ,s)= Z s

0

Z 1 0

f(re^2πiθ)

pdθdµ(r)

>

Z s 0

exp Z 1

0

log

f(re^2πiθ)

pdθdµ(r)

= Z s

0

|f(0)|^p Y

z∈Zs(f)

max r^p

|z|^p,1

dµ(r)

>

Z s 0

|f(0)|^p Y

z∈Zs(F)

max r^p

|z|^p,1

dµ(r)

=|f(0)/F(0)|^p Z s

0

exp Z 1

0

log

F re^2πiθ

p dθdµ(r).

Therefore, (2.1) |f(0)|

kfk_Ap(µ,s)

6a₀|ζ₀| Z s

0

exp Z 1

0

log

F(re^2πiθ)

p dθdµ(r) ^−1/p

.

Recall that for each r and θ, F(re^2πiθ) is a Gaussian random variable with the same distribution aska^(r)k2ζ0, where a^(r)n :=anrⁿ. Write

Gr(θ) :=F(re^2πiθ)/ka^(r)k2,

so that G_r(θ) is a standard complex Gaussian random variable for each rand θ. Hölder’s inequality and the arithmetic mean-geometric mean in- equality yield

(2.2) Z s

0

exp Z 1

0

log

F(re^2πiθ)

p dθdµ(r) ^−1/p

= Z s

0

ka^(r)k^p₂exp Z 1

0

log|Gr(θ)|^pdθdµ(r) ^−1/p

6 Rs

0 ka^(r)k^p₂expR1

0 log|Gr(θ)|⁻¹dθdµ(r) Rs

0 ka^(r)k^p₂dµ(r)^1+1/p 6

Rs

0 ka^(r)k^p₂R1

0 |Gr(θ)|⁻¹dθdµ(r) Rs

0ka^(r)k^p₂dµ(r)^1+1/p .

Multiplying both sides bya₀|ζ0|and using (2.1), we have (2.3) |f(0)|

kfk_Ap(µ,s)

6 a0|ζ0| ·Rs

0ka^(r)k^p₂R1

0 |Gr(θ)|⁻¹dθdµ(r) Rs

0ka^(r)k^p₂dµ(r)^1+1/p .

Recall that for each r and θ, Gr(θ) and ζ0 are both standard complex Gaussians, and (G_r(θ), ζ₀) is jointly Gaussian. By a version of Slepian’s lemma due to [9], we have

E

|ζ₀| · |G_r(θ)|⁻¹

6E[|ζ₀|]·E

|G_r(θ)|⁻¹

= 1·√ π.

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 multisetW, letpW(z) :=Q

w∈W(z−w) be the monic polyno- mial whose zeros areW (with multiplicity). For any analytic function f whose zeros includeW, the functionf /pW is analytic. Therefore,

max

( |f(0)|

kfk_Ap(µ,s)

; 06≡f ∈A(µ), Z(f)⊃Zs(F) )

= max

( |f(0)p_Zs(F)(0)|

f p_{Zs(F) Ap(µ,s)}

; 06≡f ∈A(µ) )

.

Restricting to polynomials f with rational coefficients, we see that this maximum is measurable provided p_Zs(F) is measurable. Now there is a measurable set (of probability 0) where lim sup|ζn|^1/n>1; off of this set, Z_s(F) is finite and can be determined by looking at the values ofF 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 thatE

|Gr(θ)|⁻¹

=√

π and|ζ0|<∞to obtain E

"Z s

0

exp Z 1

0

log

F(re^2πiθ)

p dθdµ(r) ^−1/p#

6

√π Rs

0ka^(r)k^p₂dµ(r)^1/p. Ass↑r_µ, the right-hand side tends to 0, which already gives Theorem 1.1 via (2.1).

Proof of Theorem 1.6. — We may assume thata₀ 6= 0. Suppose that 0∈/Z(f)⊇Z(F). As before, we have for 0< s < rµ

|f(0)|

kfk_Ap/N(µ,s)

!^1/N

6|a^N₀ ζ₀^N +b(0)|^1/N· Z s

0

exp Z 1

0

log

F(re^2πiθ)^N+b(re^2πiθ)

p/Ndθdµ(r) ^−1/p

. Write

Hr(θ) :=

F(re^2πiθ)^N +b(re^2πiθ)

/ka^(r)k^N₂. In the same way as before, we obtain

(2.4) Z s

0

exp Z 1

0

log|F(re^2πiθ)^N+b(re^2πiθ)|^p/Ndθdµ(r) ^−1/p

= Z s

0

ka^(r)k^p₂exp Z 1

0

logHr(θ)^p/Ndθdµ(r) ^−1/p

6 Rs

0 ka^(r)k^p₂expR1

0 logHr(θ)^−1/Ndθdµ(r) Rs

0ka^(r)k^p₂dµ(r)^1+1/p 6

Rs

0 ka^(r)k^p₂R1

0 Hr(θ)^−1/Ndθdµ(r) Rs

0 ka^(r)k^p₂dµ(r)^1+1/p . We have for anyβ that

E

H_r(θ)^−β

=Eh

|ζ₀^N +b(re^2πiθ)/ka^(r)k^N₂|^−βi .

Nowζ₀^N has densityρ:z7→c|z|^{−2(N−1)/N}e^−|z|2 (with respect to area mea- sureλ₂, for some constantc) that is decreasing in|z|. Therefore, given any α∈C, the rearrangement inequality of Hardy and Littlewood yields

P

|ζ₀^N|< r

= Z

|z|<r

ρ(z) dλ2(z)>

Z

|z−α|<r

ρ(z) dλ2(z)

=P

|ζ₀^N−α|< r ,

which is to say that|ζ₀^N|is stochastically dominated by|ζ₀^N−α|. Thus, for all 0< β <2/N,

E

|ζ₀^N −α|^−β 6E

|ζ₀^N|^−β

= Z ∞

0

e^−t

t^βN/2dt= Γ(1−βN/2).

Therefore,

(2.5) E

Hr(θ)^−β

6Γ(1−βN/2).

Multiply both sides of the inequality (2.4) by|a^N₀ ζ₀^N +b(0)|^1/N and use Hölder’s inequality to bound the resulting expectation:

Eh

|a^N₀ζ₀^N+b(0)|^1/N·Hr(θ)^−1/Ni 6E

|a^N₀ζ₀^N+b(0)|⁴4N¹ Eh

H_r(θ)^{−4/(4N−1)}i^4N−1_4N 6 a^4N₀ (2N)! + 4|b(0)|²a^2N₀ N! +|b(0)|^41/4N

Γ

2N−1 4N−1

^4N−1_4N ,

where in the last inequality, we used (2.5) withβ := 4/(4N−1).

Proof of Theorem 1.7. — We established (1.3) during the proof of The- orem 1.3, where we rely on (1.8) and the fact thatE[|ζ0|^p] = Γ(1 +p/2) for an equivalent expression on the right-hand side. That (2) implies (3) follows from the arithmetic mean-geometric mean inequality. That (3) implies (1) and (1) implies (4) are obvious. That (4) implies (2) 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. — LetV be a separable topological vector space. Let φ:V →[0,∞]be Borel measurable, c∈[1,∞), andc₁, c₂∈(1,∞)satisfy for all x, y ∈ V that φ(−x) = φ(x), c2φ(x) 6 φ(√

2x) 6 c1φ(x), and 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 thatE

e^αφ(X)β

<∞.

Proof. — Suppose that φ (x−y)/√ 2

6 τ and φ (x+y)/√ 2

> t.

Thenφ(x−y)6c₁τ andφ(x+y)> c₂t. Also,φ(2y) =φ(√

2²y)6c²₁φ(y), whenceφ(x+y)6cφ(x−y) +cc²₁φ(y). Thereforeφ(y)>(c2t−cc1τ)/(cc²₁).

Symmetry gives the same lower bound onφ(x). It follows that (A.1) P[φ(X)6τ]P[φ(Y)> t]

=Ph φ

(X−Y)/√ 2

6τ, φ

(X+Y)/√ 2

> ti 6P

φ(X)>(c2t−cc1τ)/(cc²₁)² .

Chooseτ <∞ so thatP[φ(X)6τ]>e/(1 +e). Define recursively t₀ :=

(cc1/c2)τ >τ andtn+1:= (cc²₁/c2)tn+t0; thus, tn =(cc²₁/c₂)ⁿ⁺¹−1

cc²₁/c2−1 t0< c3(cc²₁/c2)ⁿt0

for some constantc3<∞. The display (A.1) yields P[φ(X)> tn+1] =P[φ(Y)> tn+1]61 +e

e P[φ(X)> tn]², whence if we writey_n := ^1+e_e P[φ(X)> t_n], theny_n+1 6y_n², and so y_n 6 y²₀ⁿ 6e⁻²ⁿ. Therefore,

P

φ(X)> c₃(cc²₁/c₂)ⁿt₀

6e⁻²ⁿ=e^{−(cc21/c2)βn}, whereβ:= log 2/log(cc²₁/c2)>0. This means that

P[φ(X)> t]6e^−c4tβ

for somec4 > 0 and all t > t0. With α := c4/2, the conclusion may be

verified via integration by parts.

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