Riesz projection and bounded mean oscillation for Dirichlet series

arXiv:2005.11951v3 [math.FA] 28 Jun 2021

SERGEI KONYAGIN, HERVÉ QUEFFÉLEC, EERO SAKSMAN, AND KRISTIAN SEIP

A^BSTRACT. We prove that the norm of the Riesz projection fromL^∞(Tⁿ) toL^p(Tⁿ) is 1 for all n≥1 only ifp≤2, thus solving a problem posed by Marzo and Seip in 2011. This shows that H^p(T^∞) does not contain the dual space ofH¹(T^∞) for anyp>2. We then note that the dual of H¹(T^∞) contains, via the Bohr lift, the space of Dirichlet series in BMOA of the right half-plane.

We give several conditions showing how this BMOA space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection onT, we compute itsL^pnorm when 1<p< ∞, and we use this result to show that theL^∞norm of the Nth partial sum of a bounded Dirichlet series overd-smooth numbers is of order loglogN.

1. INTRODUCTION

This paper is concerned with two different ways of transferring Riesz projection to the infinite-dimensional setting of Dirichlet series: first, by lifting it in a multiplicative way to the infinite-dimensional torusT^∞and second, by using one-dimensional Riesz projection to study the partial sum operator acting on Dirichlet series. In either case, we will be interested in studying the action of the operator in question on functions inL^porH^p spaces.

By Fefferman’s duality theorem [18], Riesz projectionP₁⁺ on the unit circleT, formally de- fined as

P₁⁺¡ X

k∈Z

c_kz^k¢ :=X

k≥0

c_kz^k,

mapsL^∞(T) into and onto BMOA(T), i.e., the space of analytic functions of bounded mean oscillation. We may thus think of the image of L^∞(T^∞) under Riesz projection on T^∞ (or equivalently, in view of the Hahn–Banach theorem, the dual space H¹(T^∞)^∗) as a possible infinite-dimensional counterpart to BMOA(T). This brings us to the second main topic of this paper which is to describe some of the main properties of this space.

Our main result, given in Section 2, verifies that Riesz projection does not mapL^∞(T^∞) into H^p(T^∞) for anyp>2, whenceH¹(T^∞)^∗is not embedded inH^p(T^∞) for anyp>2. This result solves a problem posed in [38] and contrasts the familiar inclusion of BMOA(T) inH^p(T) for everyp< ∞. The key idea of the proof is to first show that the norm of a Fourier multiplier MχA :L^p(Tⁿ)→L^q(Tⁿ) corresponding to a bounded convex domainA inRⁿis dominated by

2010Mathematics Subject Classification. 30B50, 42B05, 42B30, 30H30, 30H35.

Key words and phrases. Dirichlet series, boundary behaviour.

Konyagin was supported from a grant to the Steklov International Mathematical Center in the framework of the national project ”Science” of the Russian Federation, Queffélec was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), Saksman was supported in part by the Finnish Academy grant 1309940, and Seip was supported in part by the Research Council of Norway grant 275113.

1

the norm of the Riesz projection onT^n+m formsufficiently large, depending on A. Another crucial ingredient is Babenko’s well-known lower estimate for spherical Lebesgue constants.

We then proceed to view H¹(T^∞)^∗ as a space of Dirichlet series, employing as usual the Bohr lift. This leads us in Section 3 to a distinguished subspace ofH¹(T^∞)^∗which is indeed a

“true” BMO space, namely the family of Dirichlet series that belong to BMOA of the right half- plane. By analogy with classical results onT, we give several conditions for membership in this space, also for randomized Dirichlet series, and we describe how this BMOA space relates to some other function spaces of Dirichlet series.

In Section 4, we study Dirichlet polynomials of fixed lengthNand compare the size of their norms inH^p, BMOA, and the Bloch space. One of these results is then applied in the final Section 5, where we turn to our second usage of Riesz projection. Here we present an explicit device for expressing theNth partial sum of a Dirichlet series in terms of one-dimensional Riesz projection and give someL^p estimates for the associated partial sum operator.

We refer the reader to [23] and [41] (see especially [41, Section 6]) for definitions and basics on Hardy spaces of Dirichlet series of Hardy spaces onT^∞.

Notation. We will use the notation f(x)≪ g(x) if there is some constantC >0 such that

|f(x)| ≤C|g(x)|for all (appropriate)x. If we have both f(x)≪g(x) andg(x)≪f(x), then we will writef(x)≍g(x). If lim_x→∞f(x)/g(x)=1, then we writef(x)∼g(x).

Acknowledgements. We thank Ole Fredrik Brevig for allowing us to include an unpublished argument of his in this paper. We are also grateful to the referees for a number of valuable comments that helped improve the presentation.

2. THE NORM OF THERIESZ PROJECTION FROML^∞(Tⁿ)TOL^p(Tⁿ)

The normkfk^p of a function f inL^p(T^∞) is computed with respect to Haar measurem_∞ onT^∞, which is the countable product of one-dimensional normalized Lebesgue measures onT. We denote bym_nthe measure onTⁿ that is then-fold product of the normalised one- dimensional measures, andL^p(Tⁿ) is defined with respect to this measure.

We write the Fourier series of a functionf inL¹(Tⁿ) on then-torusTⁿas

(2.1) f(ζ)= X

α∈Zⁿ

fˆ(α)ζ^α.

For a functionf inL¹(T^∞) the Fourier series takes the formf(ζ)=P

α∈Z^∞_{f i n} fˆ(α)ζ^α, whereZ^∞_{f i n} stands for infinite multi-indices such that all but finitely many indices are zero. We also set Z₊:={0,1,...} so thatZⁿ₊ (respectivelyZ^∞₊) is the positive cone inZⁿ (respectivelyZ^∞). The operator

P_n⁺f(ζ) := X

α∈Zⁿ₊

fˆ(α)ζ^α

is the Riesz projection onTⁿ, and, as an operator onL²(Tⁿ), it has norm 1. If we instead view P_n⁺as an operator onL^p(Tⁿ) for 1<p< ∞, then a theorem of Hollenbeck and Verbitsky [28]

asserts that its norm equals (sin(π/p))⁻ⁿ. In an analogous way we denote by P_∞⁺ the Riesz projection onT^∞, and obviouslyP_∞⁺ is bounded onL^p(T^∞) only for p =2, when its norm equals 1.

Using this normalization, we letkP_n⁺kq,p denote the norm of the operatorP_n⁺: L^q(Tⁿ)→ L^p(Tⁿ) for q ≥p. By Hölder’s inequality, p→ kP_n⁺k∞,p is a continuous and nondecreasing function, and obviouslykP_n⁺k∞,p≤(sin(π/p))⁻ⁿ. Consider the quantity

p_n:=sup©

p≥2 :kP_n⁺k∞,p≤1ª ,

which we following [20] call the critical exponent ofP_n⁺. The critical exponent is well-defined since clearlykP_n⁺k∞,2=1. We also set

p_∞:=sup©

p≥2 :kP_∞⁺k∞,p≤1ª .

Defining A_mf(z1,z₂,...) := f(z1,...,z_m,0,0,...) and using thatkA_mfkp → kfkp asm→ ∞for every f inL^p(T^∞) and 1≤p≤ ∞, we see that in fact

p_∞= lim

n→∞p_n. This also follows from the proof of Theorem 2.1 below.

Marzo and Seip [38] proved that the critical exponent ofP₁⁺is 4 and moreover that 2+2/(2ⁿ−1)≤p_n<3.67632

for n >1. Recently, Brevig [10] showed that limn→∞p_n ≤3.31138. The following theorem settles the asymptotic behavior of the critical exponent ofP_n⁺whenn→ ∞.

Theorem 2.1. We have p_∞=limn→∞p_n=2.

By considering a product of functions in disjoint variables, we obtain the following im- mediate consequence concerning the Riesz projectionP_∞⁺ on the infinite-dimensional torus, formally defined as

P_∞⁺³ X

k∈Z^(∞)

cαz^α´

:= X

α∈N^(∞)

cαz^α.

Corollary 2.2. The Riesz projection P_∞⁺ is not bounded from L^qto L^p when2<p<q≤ ∞. In turn, since the “analytic” dual ofH¹obviously equalsP_∞⁺(L^∞(T^∞)), we obtain a further interesting consequence.

Corollary 2.3. The dual space H¹(T^∞)^∗is not contained in H^p(T^∞)for any p>2.

The latter result has an immediate translation in terms of Hardy spaces of Dirichlet series, as will be recorded in Corollary 3.1 below.

The proof of Theorem 2.1 deals with the (pre)dual operatorP_∞⁺ :L^q(Tⁿ)→L¹(Tⁿ), where q <2. The idea is to prove first that for the characteristic functionχ_A of a bounded convex domainAinRⁿ, the norm of the Fourier multiplierM_χA onTⁿis actually bounded by that of P_n⁺_+m for large enoughm, depending on A. This key observation will be applied whenA is a large ballB(0,R) inRⁿ, and the desired result is deduced by invoking the following result of Ilyin [29].

Theorem 2.4. The circular Dirichlet kernel

D_R,n(ζ) := X

α∈Zⁿ:kαk≤R

ζ^α

onTⁿ satisfieskD_R,nkL¹(Tⁿ) ≥cR^(n−1)/2, where c =c(n)>0and k · kstands for the standard Euclidean norm.

Babenko’s famous 1971 preprint (see [3, 37]) gives another proof. Moreover, it establishes a comparable upper bound, which can also be found in Ilyin and Alimov’s paper [1]. We refer to Liflyand’s review [36] for further information on the related literature and for a simple proof of Theorem 2.4.

Proof of Theorem 2.1. Fixn≥2 andα=(α1,...,α_n)∈Zⁿtogether withβ^j ∈Zⁿandb_j ∈Zfor j =1,...,m, wherem∈Nis also fixed. We considern+mlinear functionsφ_j :Zⁿ→Z, with j=1,...,n+m, where

φ_j(α) :=α_j, j=1,...,n,

φ_n+j(α) :=(α,β^j)+b_j, j=1,...,m.

We associate with any trigonometric polynomialf as in (2.1) (that is, anyf of the form (2.1) with finitely many non-zero terms) the function

g(η) := X

α∈Zⁿ

fˆ(α)

nY+m j=1

η^φ_j^j(α), whereη=(η1,...,η_n+m)∈T^n+m.

Lemma 2.5. We havekgkp= kfkp for0<p≤ ∞. Proof. Set

η^′:=(η₁,...,η_n), η^′′:=(η_n+1,...,η_n+m).

We have

g(η)=ψ₀(η^′′) X

α∈Zⁿ

fˆ(α) Yn j=1

(ψj(η^′′)ηj)^αj, where

ψ₀(η^′′) := Ym k=1

η^b_n^k

+k, and ψj(η^′′) := Ym k=1

η^β

k j

n+k for j=1,...,n.

We clearly haveψ_j(η^′′)∈Tforj =0,...,n. For a fixedη^′′inT^mconsiderg as a function ofη^′: g(η)=g_η′′(η^′).

Set ˜η^′=(˜η₁,..., ˜η_n), where ˜η_j=ψ_j(η^′′)η_j for j=1,...,n. We see that g_η′′(η^′)=ψ₀(η^′′)f(˜η^′).

We therefore obtain the asserted isometry:

kg_η′′kp= kfkp

By duality, for any positive integer N andp >2, we havekP_N⁺k∞,p = kP_N⁺kp^′,1 wherep^′= p/(p−1). Hence, to prove Theorem 2.1, we have to show that for anyq in (1,2) there exist a positive integerNandg inL^q(T^N) such that

(2.2) kgkq=1, kP_N⁺gk1>1.

Indeed, by duality, this will imply the existence of a functionhinL^∞(Tⁿ) such that khk∞=1, kP⁺_N(h)kq^′>1,

whereq^′=q/(q−1). Sinceq<2 is arbitrary, Theorem 2.1 then follows.

For a bounded set E inRⁿ and a function f in L¹(Tⁿ), we consider a partial sum of the Fourier series off:

¡S_Ef¢

(ζ) := X

α∈E∩Zⁿ

fˆ(α)ζ^α.

Note that as an operator,S_E coincides with the Fourier multiplierM_χE. We say that a polytope EinRⁿis non-degenerate if it is not contained in a hyperplane.

Lemma 2.6. Let1<q<2. Assume that there is a non-degenerate convex polytope E inRⁿwith integral vertices such that, for some f in L^q(Tⁿ)with a finite set of non-zero Fourier coefficients

fˆ(α), we have

kfkq=1, kS_E(f)k1>1.

Then there are a positive integer N∈Nand a function g in L^q(T^N)satisfying(2.2).

Proof. Lete:=(1,1,...,1)∈Z⁺_n. By considering insteadE+N eand (η1...ηn)^Nf(η) with large enoughN∈N, if necessary, we may assume thatEand the Fourier coefficients of f satisfy (2.3) E⊂Z⁺_n and fˆ(α)6=0 ⇒ αj≥0 for all j =1,...,n.

It is known thatE is the intersection of closed semispaces, bounded by the hyperplanes con- taining the faces ofE of dimensionn−1 (see [35, Ch. 1, Thm. 5.6]). All hyperplanes are de- termined by their intersections with the set of the vertices ofE. Since the vertices are integral, the semispaces can be defined by inequalities

(α,β^j)+b_j≥0, j=1,...,m, whereβ^j∈Zⁿ,b_j ∈Zfor j=1,...,m. Thus

E=

n\+m j=n+1

{α∈R^m:φj(α)≥0}, whereφ_j(α)=(α,β^j−n)+b_j−n, j=n+1,...,n+m.

We setN:=n+mand construct the functiong from f as in Lemma 2.5. Using that lemma, we get

kgkq= kfkq=1, kP_N⁺gk1= kS_E(f)k1>1,

and Lemma 2.6 follows.

To construct an integern, a polytopeE, and a functionf satisfying Lemma 2.6, we take first nsatisfying the inequality

(2.4) n>q/(2−q).

For sufficiently largeR, letE be the convex hull of the integral points contained in the eu- clidean ball {α∈Rⁿ:kαk ≤R}. Hence for any function f inL¹(Tⁿ), we have

(SEf)(ζ)= X

α∈Zⁿ:kαk≤R

fˆ(α)ζ^α. Recall the circular Dirichlet kernel from Theorem 2.4:

D_R,n(ζ)= X

α∈Zⁿ:kαk≤R

ζ^α. Define the functionfe(ζ) := X

|α1|≤R

··· X

|αn|≤R

ζ^αso thatSRfe=D_R,n. It is easy to see that

kfek^q=

°°

°°

° X

|α1|≤R

ζ^α₁¹

°°

°°

°

n

q

≤C R^n(1−1/q),

whereC =C(q,n)>0. In view of (2.4), which amounts to ⁿ⁻₂¹ >n(1−_q¹), and by recalling Theorem 2.4, we obtain

kS_E(fe)k1> kf˜kq. for sufficiently largeR. Taking f := fe±

kfekq, we get a function f satisfying the conditions of

Lemma 2.6, and this completes the proof of Theorem 2.1.

3. THE SPACE OFDIRICHLET SERIES INBMOA

The result of the preceding section is purely multiplicative in the sense that it only involves analysis on the product spaceTⁿ. Function spaces onTⁿor onT^∞may however, by a device known as the Bohr lift (see below for details), also be viewed as spaces of Dirichlet series. From an abstract point of view (see for example [42, Ch. 8]), this means that we equip our function spaces with an additive structure that reflects the additive order of the multiplicative group of positive rational numbersQ₊. This results in interesting interaction between function theory in polydiscs and half-planes that sometimes involves nontrivial number theory.

As we will see in the next subsection, this point of view leads us naturally fromH¹(T^∞)^∗ to the space of ordinary Dirichlet seriesP_∞

n=1ann^−s that belong to BMOA, i.e., the space of analytic functionsf(s) in the right half-plane Res>0 satisfying

(3.1) sup

σ>0

Z_∞

−∞

|f(σ+i t)|²

1+σ²+t²d t < ∞ and

kfkBMO:=sup

I⊂R

1

|I| Z

I

¯¯

¯¯f(i t)− 1

|I| Z

I

f(iτ)dτ

¯¯

¯¯d t< ∞.

Here the supremum is taken over all finite intervalsI; (3.1) means thatg(s) :=f(s)/(s+1) be- longs to the Hardy spaceH²(C0) of the right half-planeC₀, and thenf(i t) :=limσ→0⁺f(σ+i t) exists for almost all realtby Fatou’s theorem applied tog. We will use the notation BMOA∩D

for this BMOA space, whereD is the class of functions expressible as a convergent Dirichlet series in some half-plane Res>σ₀.

The space BMOA∩Darose naturally in a recent study of multiplicative Volterra operators [11]. We refer to that paper for a complementary discussion of bounded mean oscillation in the context of Dirichlet series. By combining [11, Cor. 6.4] and [11, Thm. 5.3], we may conclude that BMOA∩D can be viewed, via the Bohr lift, as a subspace ofH¹(T^∞)^∗. This inclusion may however be proved in a direct way by an argument that we will present in the next subsection.

3.1. The Bohr lift and the inclusionBMOA∩D ⊂(H¹)^∗. We begin by considering an ordi- nary Dirichlet series of the form

(3.2) f(s)=

X∞ n=1

a_nn^−s.

By the transformationz_j =p⁻_j^s(herep_j is thejth prime number) and the fundamental theo- rem of arithmetic, we have the Bohr correspondence,

(3.3) f(s) :=

X∞ n=1

ann^−s ←→ Bf(z) := X∞ n=1

anz^κ(n),

whereκ(n)=(κ1,...,κ_j,0,0,...) is the multi-index such thatn=p₁^κ1···p^κ_j^j. The transforma- tionBis known as the Bohr lift. For 0<p< ∞, we defineH^p as the space of Dirichlet series

f such thatBf is inH^p(T^∞), and we set kfk^Hp:= kBfkH^p(T^∞)=

µZ

T^∞|Bf(z)|^pd m_∞(z)

¶_p¹ . Note that forp=2, we have

kfk^H2= µX_∞

n=1|a_n|²

¶¹₂ . In terms of the spacesH^p, Corollary 2.3 takes the form

Corollary 3.1. The dual space(H ¹₎∗is not contained inH ^p _{for any p>2.}

We will now use the notationC_θ :={s =σ+i t :σ>θ}. The conformally invariant Hardy spaceH_i^p(Cθ) consists of functionsf that are analytic onC_θand satisfy

kfkH_i^p(C_θ):=sup

σ>θ

µ1 π

Z

R|f(σ+i t)|^p d t 1+t²

¶_p¹

< ∞.

These spaces show up naturally in our discussion in the following two ways. First, we will repeatedly use that a functiong analytic onC₀is in BMOA if and only if the measure

dµ(s) := |g^′(σ+i t)|²σdσ d t 1+t²

is a Carleson measure forH_i¹(C₀), which means that there is a constantC such Z

C₀|f(s)|dµ(s)≤CkfkH_i¹(C0)

for all f inH_i¹(C0). The smallest such constantC is called the Carleson norm of the measure.

Second, by Fubini’s theorem, we have the following connection betweenH^pandH_i^p(C0):

(3.4) °°f°°^p_Hp=

Z

T^∞kf_χk^p_H^p

i (C0)d m_∞(χ),

whereχis a character onQ⁺, i.e., a completely multiplicative function taking only unimodular values, and

fχ(s) := X∞ n=1

χ(n)ann^−s.

Here we recall that an arithmetic functiong :N→Cis completely multiplicative if it satisfies g(nm)=g(n)g(m) for all integersm,n≥1. A completely multiplicative functiong satisfies g(1)=1 unless g vanishes identically, and it is completely determined by its values at the primes.

Note that we identify via the Bohr liftα7→p^αthe groupZ^(∞)with the groupQ⁺, and by dual- ity the groupT^∞with the group of completely multiplicative functionsχ:N→T. Accordingly, we identify the Haar measuresd m_∞(z) andd m_∞(χ) of both groups. We also used in (3.4) the fact that, for almost every characterχandP_∞

n=1a_nn^−sinH^p, the seriesP_∞

n=1a_nχ(n)n^−scon- vergesm_∞-almost everywhere inC₀, and defines an element inH_i^p(C₀). For these facts, we refer e.g. to [23, Section 4.2] and [5, Thm 5].

From (3.4) we may deduce Littlewood–Paley type expressions for the norms ofH^p. This was first done for p =2 in [5, Prop. 4], and later for 0<p< ∞ in [7, Thm. 5.1], where the formula

(3.5) kfkH^{p p}≍ |f(+∞)|^p+4 π

Z

T^∞

Z

R

Z_∞

0 |f_χ(σ+i t)|^p−2|f_χ^′(σ+i t)|²σdσ d t

1+t²d m_∞(χ) was obtained. Whenp=2, we have equality between the two sides of (3.5).

The Littlewood–Paley formula (3.5) forp=2 may be polarized, so that we have

〈f,g〉^H2=f(+∞)g(+∞)+4 π

Z

T^∞

Z

R

Z_∞

0 f_χ^′(σ+i t)g_χ^′(σ+i t)σdσ d t

1+t²d m_∞(χ).

Hence, by the Cauchy–Schwarz inequality and (3.5), we have for f inH¹andg in BMOA∩D,

¯¯〈f,g〉^H2−f(+∞)g(+∞)¯¯²≤ 4 π

Z

T^∞

Z

R

Z_∞

0 |f_χ(σ+i t)|⁻¹|f_χ^′(σ+i t)|²σdσ d t

1+t²d m_∞(χ)

× Z

T^∞

Z

R

Z_∞

0 |f_χ(σ+i t)||g_χ^′(σ+i t)|²σdσ d t

1+t²d m_∞(χ)

≪ kfk^H1 Z

T^∞kf_χkH_i¹(C0)d m_∞(χ)= kfk²H1,

where we in the second step used the Littlewood–Paley formula forp=1 and that

|g_χ^′(σ+i t)|²σdσ d t 1+t²

is a Carleson measure forH_i¹(C₀), with Carleson constant uniformly bounded inχ, as follows from [11, Lem. 2.1 (ii) and Lem. 2.2]. Hence we conclude that a Dirichlet seriesg in BMOA∩D belongs to (H¹)^∗.

The “reverse” problem of finding an embedding of (H¹)^∗into a “natural” space of functions analytic inC_1/2appears challenging. (This is a reverse question only in a rather loose sense as we are now considering functions defined inC_1/2.) It was mentioned in [43, Quest. 4] that (H¹)^∗ is not contained in H_i^q(C1/2) for anyq >4. Since no argument for this assertion was given in [43], we take this opportunity to offer a proof¹. To begin with, let us consider the interval from 1/2−i to 1/2+i and let E denote the corresponding local embedding ofH ² intoL²(−1,1), given byE f(t) :=f(1/2+i t), so that

kE fk²_L²_(−1,1)= Z1

−1|f(1/2+i t)|²d t. Then the adjointE^∗: L²(−1,1)→H ²_is

E^∗g(s) := X∞ n=1

b g(logn)

pn n^−s, wheregb(ξ)=R1

−1e^−iξtg(t)d t. Fix 0<β<1 and setg_β(t) := |t|^β−1. Plainly,g_βis inL^q(−1,1) if and only ifβ>1−1/q. Moreover, ifξ≥δ>0, thengbβ(ξ)≍ξ^−β, where the implied constants depend only onδandβ. We now invoke Helson’s inequality [25, p. 89]

°°

° X∞ n=1

a_nn^−s

°°

°₁≥ µX_∞

n=1

|an|² d(n)

¶^1/2 ,

whered(n) is the divisor function. We then use the classical fact thatP

n≤x1/d(n) is of size x(logx)^−1/2; the precise asymptotics of this summatory function was first computed by Wilson [47, Formula (3.10)] and may now be obtained as a simple consequence of a general formula of Selberg [44]. Takingβ=1/4, we may therefore infer by partial summation thatE^∗is un- bounded fromL^q(−1,1) toH¹_wheneverq<4/3. By duality we conclude that for anyq>4, there areϕin (H ¹₎^∗that are not locally embedded inL^q(−1,1) and hence do not belong to H_i^q(C1/2). Note that here (H ¹₎^∗is identified as a subspace ofH²(with respect to the natu- ral pairings ofL²(−1,1) andL²(T^∞)) ,whenceE^∗∗g =E g for (H ¹₎∗. In view of Corollary 2.3, it is natural to ask if the situation is even worse, namely that (H¹)^∗fails to be contained in H_i^q(C1/2) for anyq>2.

We conclude from the preceding argument that there is no simple relation between (H¹)^∗ and BMOA(C_1/2). We may further illustrate this point by the following example. The Dirichlet series

h(s) := X∞ n=2

1

lognn^−s−1/2

1We thank Ole Fredrik Brevig for showing us this argument and allowing us to include it in this paper.

belongs to BMOA(C_1/2) (see (3.7)) below), but it is unknown whether it is in (H¹)^∗. It would be interesting to settle this question about membership in (H¹)^∗, ashis both a primitive of ζ(s+1/2)−1 and the analytic symbol of the multiplicative Hilbert matrix [12].

3.2. Fefferman’s condition for membership inBMOA∩D. The following theorem gives in- teresting information about Dirichlet series in BMOA. It is an immediate consequence of existing results, as will be explained in the subsequent discussion.

Theorem 3.2. (i) Suppose that a_n≥0for every n≥1. Then f(s) :=P_∞

n=1a_nn^−sis inBMOA if and only if

(3.6) S²:=sup

x≥e

X∞ k=1

³ X

x^k≤n<x^k+1

an

´2

< ∞, and we have S≍ kfkBMOA.

(ii) If P_∞

n=1|a_n|n^−sis inBMOA, thenP_∞

n=1a_nn^−s is inBMOA.

It is immediate from (i) that

(3.7) X^∞

n=2

1

lognn^−s−1 is in BMOA (see [11, Thm. 2.5]). By Mertens’s formula

(3.8) X

p≤x

1

p =loglogx+M+O¡

(logx)⁻¹¢ , where the sum is over the primesp, part (i) also implies thatP

pp^−1−s is in BMOA, and con- sequently logζ(s+1) is a function in BMOA, whereζ(s) is now the Riemann zeta function.

Then part (ii) of Theorem 3.2 implies also thatP

pχ(p)p^−1−s is in BMOA for any sequence of unimodular numbersχ(p). In fact, we have more generally:

Corollary 3.3. A Dirichlet seriesP

pa_pp^−s over the primes p is inBMOAif and only if

(3.9) sup

x≥e

X∞ k=1

³ X

x^k≤p<x^k+1

|a_p|´2

< ∞.

Corollary 3.3 is a consequence of part (i) of Theorem 3.2 and the fact (see [11, Lem. 2.1]) that Ppa_pp^−s is in BMOA if and only ifP

pa_pχ(p)p^−s is in BMOA for every sequence of unimodu- lar numbersχ(p).

The sufficiency of condition (3.6) in Theorem 3.2)(i) follows as a corollary to anH¹multi- plier theorem of Sledd and Stegenga [46, Thm. 1] via Fefferman’s duality theorem [18, 19] and Parseval’s theorem. The necessity also follows from [46, Thm. 1] if we first note that for any f in H¹(C0), using the standardH²factorization of H¹, we may construct g inH¹(C0) with kgkH¹(C₀)= kfkH¹(C₀) andgb(ξ)≥ |fb(ξ)| ≥0 for allξ∈R. Here fb,gbrefer to the Fourier trans- forms of the boundary values on the imaginary axis. A corresponding result for BMO in the

unit disc is stated in [46, Cor. 2]: The Taylor seriesP_∞

m=0cmz^m withcm≥0 belongs to BMO of the unit circleTif and only if

sup

m≥1

X∞ j=0

Ã_m−1 X

r=0

cm j+r

!2

< ∞.

Other proofs of this result, relying more directly on Hankel operators, can be found in [9, 27].

This result is commonly known to have appeared in unpublished work of Fefferman.

To establish part (ii) of Theorem 3.2, we use the following Carleson measure characteriza- tion of BMOA∩Dwhich could be used to give an alternative proof of part (i) of Theorem 3.2.

Lemma 3.4. Suppose that f is in H_i²(C0)∩D. Then f is inBMOA∩Dif and only if there exists a positive constant C such

(3.10) sup

t∈R

Z_h

0

Z_t+h

t |f^′(σ+iτ)|²σdτdσ≤C h

for0≤h≤1. Moreover, the best constant C in ( 3.10) andkfk²_BMOare equivalent.

Proof. We first observe that (3.10) and the assumption thatf is inH_i²(C0) imply, by the max- imum modulus principle, that f^′(σ+i t) is uniformly bounded byO(p

C) forσ≥1. Then, if h>1 andt∈Rare given and

I:= Zh

0

Zt+h

t |f^′(σ+iτ)|²σdτdσ

= Z₁

0

hZ^t+h

t |f^′(σ+iτ)|²dτi σdσ+

Z_h

1

hZ^t+h

t |f^′(σ+iτ)|²dτi

σdσ=:I₁+I₂, we haveI₁≪C hby (3.10), while

I₂≪ Z_∞

1

hZ^t+h

t |f^′(σ+iτ)|²dτi

σdσ≪ Zt+h

t

hZ∞

1 σC4^−σdσi

dτ≪C h.

To obtain the final estimate above, we used thatf^′(σ+i t)=O(p

C2^−σ), which holds uniformly

intwhenσ≥1 because f is a Dirichlet series.

Part (ii) of Theorem 3.2 is immediate from this lemma along with a property of almost peri- odic functions established by Montgomery [40, p. 131] (see also [39, p. 4]) which asserts that if|a_n| ≤b_n, then for sums with a finite number of non-zero terms

ZT₁+T

T₁−T

¯¯X

ane^iλnt¯¯²d t≤3 ZT

−T

¯¯X

bne^iλnt¯¯²d t.

HereT >0,T₁is a real number,a_n,b_nrespectively complex and nonnegative coefficients, and λ_nare distinct real frequencies.

We will now apply Theorem 3.2 to see how our BMOA space of Dirichlet series relates to Hardy spaces and the Bloch space. We denote as usualH^∞(C0)∩DbyH∞, and we say that a functionf(s) analytic in Res>0 is in the Bloch spaceB_if

kfk^B:= sup

σ+i t:σ>0

σ|f^′(σ+i t)| < ∞.

We have

H^∞⊂BMOA∩D⊂ \

0<q<∞

H^q,

where the inclusion to the left is trivial and that to the right was established in [11, Lem. 2.1].

Hence, in contrast to (H¹)^∗itself, the subspace BMOA∩Dis included inT

0<q<∞H^q. More- over, is a classical fact and easy to see that BMOA⊂B_.

The following consequence of Corollary 3.3 is a Dirichlet series counterpart to a result of Campbell, Cima, and Stephenson [13] that further enunciates the relation between the spaces in question. Our proof is close to that found in [26].

Corollary 3.5. There exist Dirichlet series that belong toB_and^T_0<q<∞H^qbut not toBMOA.

Proof. It is an easy consequence of the definition of the Bloch space thatP_∞

n=1ann^−s with an≥0 is inBif and only if

(3.11) sup

x≥2

X

x≤n<x²

a_n< ∞.

Indeed, if (3.11) holds, then we use it withxj=exp(2^j/σ), xj+1=x²_j, to show that forσ>0, X

n≥2

a_nσlogn e^−σlogn≤X

j

2^−j¡ X

xj≤n<xj+1

a_n¢

≪X

j

2^−j. Conversely, ifP

n≥2a_nσlogne^−σlogn≤C for allσ>0, then choosingσ=1/logx, we see that the sum on the left-hand side of (3.11) is bounded by C e²/2. Let P_j be the primes in the interval [e^2j,e^2j+1]. Then|P_j| ∼(e−1)e^2j2^−j by the prime number theorem. Setting a_p := e^−2j2^j if p is inP_j and a_p =0 otherwise, we see from (3.11) thatP

pa_pp^−s is in the Bloch space, but from part (i) of Theorem 3.2 that it fails to be in BMOA.

We next recall Khinchin’s inequality for the Steinhaus variables Z_p (that are i.i.d. random variables with uniform distribution onT):

E¯¯X

p

apZp

¯¯^q≍¡X

p |ap|²¢q/2,

with the implied constants only depending on q >0 (see [33, Thm. 1]). Since in the Bohr correspondencep⁻_k^s corresponds to the independent variablez_k, we see that they form a se- quence of Steinhaus variables with respect to the Haar measure onT^∞. Thus, in view of the bound

X

p

a²_p≪ X∞ j=0

e^−2j2^j < ∞, Khinchin’s inequality implies thatP

pa_pp^−sbelongs toH^q.

3.3. The relation between Dirichlet series in H^∞,BMOA, andB_. We turn to some further comparisons between the three spacesH^∞, BMOA∩D, andB_∩D. We begin with a discus- sion of uniform and absolute convergence of Dirichlet series inB_∩D. The following lemma will be useful in this discussion. Here we use the notation log₊x:=max(0,logx) forx>0, and we will also write (Tcf)(s) :=f(s+c) in what follows.

Lemma 3.6. Suppose that f(s)=P_∞

n=1ann^−sis inB_∩D. Then

|a_n| ≤ekfk^B, n≥2, (3.12)

|f(σ+i t)−a₁| ≤ µ

log₊1

σ+C2^−σ

¶

kfk^B, σ>0, (3.13)

for some absolute constant C . Up to the precise value of C , these bounds are both optimal.

Proof. To prove (3.12), we use thatTεf^′is inH ^∞_{for everyε>}0. By either viewing the coef- ficients of a Dirichlet series as Fourier coefficients or using thatkfk^H2≤ kfk^H∞, we see that they are dominated by itsH^∞norm. We therefore have

|a_n|(logn)n^−ε≤ kT_εf^′k∞≤kfk^B ε and hence

|a_n| ≤n^εkfk^B εlogn .

We conclude by takingε=1/logn. In addition, we notice that the bound is optimal because kn^−sk^B=1/e.

To prove (3.13), we begin by noticing that (3.12) implies that (3.14) |f(σ+i t)−a₁| ≤

X∞

n=2|a_n|n^−σ≤e(ζ(σ)−1)kfk^B holds forσ≥2. Forσ≤2, we use that

|f(σ+i t)−a₁| ≤ |f(2+i t)−a₁| + Z2

σ kfk^Bdα α ≤

µ log1

σ+C

¶ kfk^B, where we in the final step used (3.14) withσ=2. The exampleP_∞

n=2n^−1−s/logn shows that

the inequality is optimal, up to the precise value ofC.

The pointwise bound (3.13) implies that what is known about uniform and absolute con- vergence of Dirichlet series inH^∞carries over in a painless way toB_∩D. In fact, a rather weak bound of the form

(3.15) |f(σ+i t)| ≤C(σ), σ>0,

suffices to draw such a conclusion, as will now be explained. To begin with we will assume thatC(σ) is an arbitrary positive function and later specify its required behavior asσ→0⁺.

First, by a classical theorem of Bohr [41, p. 145], a bound like (3.15) implies that the Dirich- let series of f(s) converges uniformly in every half-plane Res ≥σ₀>0. Following Bohr, we then see thatσ_u(f)≤0, whereσ_u(f) is the abscissa of uniform convergence, defined as the infimum over thoseσ₀such that the Dirichlet series off(s) converges uniformly in Res≥σ₀.

Second, as observed by Bohr, it is immediate thatσ_u(f)≤0 impliesσ_a(f)≤1/2, where σ_a(f) is the abscissa of absolute convergence of f, i.e., the infimum over thoseσ₀such that the Dirichlet series of f(s) converges absolutely in Res ≥σ₀. Thanks to more recent work