Hilbert spaces of Dirichlet series

Hilbert spaces of Dirichlet series

Joaquim Ortega-Cerd`a

Puerto de la Cruz, March 5,6,8,9, 2012

Overview

I Ordinary Dirichlet series.

I Regions of convergence.

I Definition and reproducing kernel.

I Local behaviour and the discrete Hilbert transform.

I Almost periodicity.

I Pointwise convergence.

I Bohr correspondence.

I L^p variants.

I Multipliers and applications.

I Composition operators.

I Other weights.

I Estimates of coefficients.

I Open problems.

Ordinary Dirichlet series

We will consider series of the form f(s) =

∞

X

n=1

ann^−s, wheres =σ+it ∈Cis a complex variable.

To make things interesting we assume that the series converges at least in a points =s₀.

Actually the region of convergence will be a half plane by the Theorem (Abel-Dedekind-Dirichlet)

If the seriesP

na_n has bounded partial sums and the sequence b_n is of bounded variation andlim_nb_n= 0 then the series P

na_nb_n converges.

Replacingf(s) by g(s) =f(s+s0) we may assumes0= 0 without loss of generality. Since|n^−s−(n+ 1)^−s|=O(n^−1−Re(s)), then n^−s is of bounded variation inC⁺₀ ={z ∈C; <z >0}.

Regions of convergence

Actually the convergence is uniform in a cone with vertexs₀ as in the picture. Thusf(s) is an holomorphic function in the half plane to the right ofs0.

In contrast to the power series the regions of convergence differ when we consider pointwise convergence, uniform convergence or absolute convergence.

Regions of convergence

Givenf(s) we can define (at least) three abcissaσ_C, σ_U, σ_A.

I σ_C is the smallest σ such that f(s) is convergent in C⁺σC.

I σU is the smallest σ such that f(s) converges uniformly in C⁺σU+ε for anyε >0.

I σA is the smallest σ such that f(s) converges absolutely in C⁺σA.

Clearlyσ_C ≤σ_U ≤σ_A. It is also easy thatσ_A−σ_C ≤1 (it can be anything in between 0 and 1).

Remark

The function f(s)may continue analytically in a region bigger than C⁺_σC.

Keep in mind the following example:

g(s) =X

n≥1

(−1)ⁿn^−s. g(s) = 2ζ(s)(2^−s−1/2) This is an entire function such thatσ_C = 0 andσ_A =σ_U = 1.

Regions of convergence

It remains to clarify what is the relation betweenσ_U andσ_A. Bohr studied another abscissaσB or abscissa of boundedness

I σ_B is the smallest σ such that f(s) converges at some point s and it is bounded in C⁺_σB+ε for anyε >0.

And he proved Theorem (Bohr) σ_B =σ_U.

In particular this means that whenever we have an analytic functionf that coincides with a Dirichlet series in a half planeC⁺a

and it is holomorphic and bounded up toC⁺_b with b<a, then the Dirichlet series converges tof up to C⁺_b.

Regions of convergence

Proof (Maurizi, Queff´elec).

We only need to prove thatσ_U ≤σ_B. Suppose that f(s) =P

a_nn^−s defines a bounded analytic function inC⁺₀ and let kfk_∞= sup_C⁺

0 |f(s)|. Denote by D_N(s) =D_N[f](s) =PN

n=1ann^−s. Then

Proposition (Balasubramanian, Calado, Queff´elec)

kD_Nk_∞≤ClogNkfk_∞. Now summing by parts:

N

X

n=1

a_nn^−(s+ε)=

N−1

X

n=1

D_n(s)(n^−ε−(n+ 1)^−ε) +N^−εD_N(s).

and the new series is dominated by ^Cε_nε+1^lognkfk_∞.

Regions of Convergence

Theorem

σA−σB ≤1/2 (Bohr) and1/2 is sharp (Bohnenblust-Hille).

We will prove a refined version.

Theorem (DFOOS) The series

∞

X

n=1

|a_n|n⁻¹²expn cp

lognlog logno is convergent for everyP

nann^−s bounded Dirichlet series inC⁺₀ if c <1/√

2 and it may be divergent if c >1/√ 2.

Definition and reproducing kernel

Definition

The spaceH² will be the Hilbert space of analytic functions defined by

f(s) =

∞

X

n=1

a_nn^−s withkfk² :=P

|a_n|² <∞.

The functions in the space converge inσ >1/2. In fact by Cauchy-Schwartz any function inH satisfiesσA ≤1/2.

There are function onH² that are only defined in C⁺_1/2.

The valuesf(s) determine the coefficients and of course the norm, but it will be interesting to express the norm directly in terms of the growth off(s).

Carlson’s theorem

Letf(s) =P∞

n=1a_nn^−s ∈ H², then for any σ > σ_U we have X

n

|a_n|²n^−2σ = lim

T→∞

1 T

Z T 0

|f(σ+it)|²dt. In particularH² is the closure of the Dirichlet polynomials p(s) =PN

n=1a_nn^−s under the norm kpk² = lim

T→∞

1 T

Z _T

0

|p(it)|²dt.

This way of computing the norm is not valid for allf ∈ H² because in generalf is not defined in the imaginary axis.

The reproducing kernel

Definition

Sincen^−s is an orthonormal basis and point evaluation is bounded forσ >1/2, the function

∞

X

n=1

n^−sn^−w¯ =ζ(s + ¯w)

is the reproducing kernel forH², i.e, for any w ∈C⁺_1/2 f(w) =hf(s), ζ(s+ ¯w)i

and the norm of the pointwise evaluation isp

k(s,s) =p ζ(2σ) which blows up as 1/√

2σ as we approach the boundary ofC1/2.

Local behaviour

Theorem

For any f ∈ Hthe following holds:

Z T 0

|f(σ+it)|²dt ≤C(T)kfk²_H ∀σ ≥1/2.

Proof: LetD(s) =PN

n=1ann^−s. The statement follows from:

Z T 0

|D(it)|²dt =T

N

X

n=1

|a_n|²+O(

N

X

n=1

n|a_n|²), which trivially implies

Z T 0

|D(σ+it)|²dt =T

N

X

n=1

|a_n|² n^2σ +O(

N

X

n=1

n

n^2σ|a_n|²).

Local behaviour

R_T

0 |D(it)|²dt =TPN

n=1|a_n|²+O(PN

n=1n|a_n|²), Z T

0

|D(it)|²dt =

N

X

n,m=1

an¯am

Z T 0

e^{i(logn−logm)t}dt =

=T

N

X

n

|a_n|²+iX

n6=m

a_n¯a_m

logn−logm −iX

n6=m

a_ne^ilognT¯a_me^ilogmT logn−logm . The last two terms are similar and can be controlled if we prove:

X

m6=n

xmyn

λm−λn

2

≤CX

m

|x_m|² dm

X

n

|y_n|² dn

,

whereδ_n= minm6=n|λ_m−λ_n|. In our caseλ_n= lognandd_n= 1/n

Local behaviour and the discrete Hilbert transform

In order to prove:

X

m6=n

x_my_n λm−λn

2

≤CX

m

|x_m|² dm

X

n

|y_n|² dn

,

consider the measureµ=P

nd_nδ_λn. The pointsλ_n∈R thus the curvature ofµ= 0 and by the definition of d_n it has linear growth (it is doubling).

ThusC(f)(z) =R f(w)

z−wdµ(w) is bounded fromL²(dµ) to L²(dµ).

Therefore there is a constantC >0 such that for allan,bm we have

X

m6=n

a_md_m λm−λn

b¯_nd_n

2

≤CX

m

|a_m|²d_mX

n

|b_n|²d_n, Finally takinga_m=x_m/d_m andb_n= ¯y_n/d_n we get the result.

Immediate consequences

This gives several immediate consequences:

Proposition If f ∈ H² then

I g(s) =f(s)/s ∈H²(C⁺_1/2).

I The zeros of f satisfy the Blaschke condition on the halfplane C⁺_1/2.

I For almost every t ∈R there exists the non-tangential limit of f at the point1/2 +it

And some not so immediate Proposition

A compactly supported measureµ is a Carleson measure for H² (i.e. R

C⁺_1/2|f|²dµ≤Ckfk² for all f ∈ H²) if and only ifµis a Carleson measure for the Hardy spaceH²(C⁺_1/2).

Almost periodicity

Letf be holomorphic inC⁺_a. Letε >0. A number τ is an translation number off if

sup

s∈C⁺a

|f(s+iτ)−f(s)| ≤ε.

Definition

A function f is uniformly almost periodic inC⁺_a if, for every ε >0, there is an M>0 such that every interval of length M contains at least onetranslation number of f .

One can prove that:

Theorem

Suppose that f(s)is a Dirichlet series and a> σ_U, then f is uniformly almost periodic inC⁺a.

Almost periodicity

This is a consequence of Kronecker’s theorem that says Theorem

Let p1 = 2,p2 = 3,p3 = 5, . . . be the primes, then {(tlogp1,tlogp2, . . . ,tlogpn)⊂Tⁿ,t∈R} is dense inTⁿ (the flow is equidistributed).

Thus for anyε >0 we can find a sequence T_n→ ∞ relatively dense such thatd(Tnlogpj,2πZ)< εfor all j = 1, . . . ,n and therefore|d_n(s−iTn)−dn(s)| ≤Cεfor a given Dirichlet polynomial of degreen.

By uniform approximation of the functionf by Dirichlet polynomials inC⁺a we get the desired result.

In particular iff ∈ H² andf(a) = 0 <(a)>1/2, then in any strip

<(a)−ε <<f <<(a) +εwe can find infinitely many zeros.

Pointwise convergence

Theorem For anyP

a_nn^−s ∈ H, the seriesP

a_nn^−s converges at s = 1/2 +it for almost every t ∈R.

Proof.

Letf : [1,∞)→Cbe defined by f(x) =a_n ifn≤x<n+ 1 and n= 1,2, . . .. Now, defineg : [0,∞)→Cas g(y) =f(e^y)e^y/2. We haveg ∈L²(R⁺) and using Carleson’s theorem for integrals we get that

Z ∞ 0

g(y)e^itydy = lim

A→∞

Z A 0

g(y)e^itydy exists a.e.int.Thus

N→∞lim Z N

1

f(x)

√x e^itlogxdx.

exists a.e.int and unwinding the definition off we get the result.

Interpolating sequences

The techniques so far were from harmonic analysis. We can get more information in terms of the behaviour of the reproducing kernel. Observe that there is an entire functionh such that

ζ(s+ ¯w) = 1

s+ ¯w+ 1+h(s+ ¯w).

This is innocent looking, but _{s+ ¯w+1}¹ is the reproducing kernel of H²(C⁺_1/2) andζ(s+ ¯w) is the reproducing kernel forH.

From this relation Olsen and Seip proved:

Theorem

Let S be a bounded sequence inC⁺_1/2. Then S is an interpolating sequence forH if and only if S is an interpolating sequence for H²(C⁺_1/2).

A sequenceS is interpolating if for any sequence of values aj/kk_ajk ∈`2 there is a functionf ∈ Hsuch that f(sj) =aj. Similarly forH²

Interpolating sequences

The two key ingredients are the local embedding ofH² in H²(C_1/2) and the dual formulation of interpolating sequence:

Lemma

A sequence S is interpolating if kX

c_j k_sj

kk_sjkk² ≥c(X

|c_j|²).

Then by a perturbative argument due to the nature of the reproducing kernels one can transfer the interpolating problem from one space to the other.

Local behaviour, again

The following theorem provides a precise description of the local behaviour ofH²:

Theorem (Olsen and Saksman)

Let f ∈H²(C⁺_1/2). Then there is an F ∈ H² such that f −F extends holomorphically through I = [1/2−i,1/2 +i].

To prove this we will first prove thatR :`²(Z\ {0})→L²[−1,1] is onto where

R(a_n) =χ_[−1,1](t)

∞

X

n=1

a_nn^−it+a−nn^it

√n

Then the theorem follows because for anyf ∈H²(C⁺_1/2) there are a_n such that R(a_n) =χ_[−1,1](t)<f. We may assume thata_n∈R. We defineF(s) = 2P

n≥1a_nn^−s and it satisfies

(<F − <f)(1/2−it) = 0, t∈[−1,1], and the theorem follows.

Local behaviour, again

In order to check the exhaustivity ofR we computeR^∗ R^∗(g) = (. . . ,gˆ(−log 2)

√2 ,gˆ(0),gˆ(0),gˆ(−log 2)

√2 , . . .) NowR is onto ifR^∗ is bounded below, i.e if for allg ∈L²([−1,1]) we have

kgk² ≤CX

n≥1

|ˆg(±logn)|²

n .

In other words for allf ∈PW we should have kfk² ≤CX

n≥1

|f(±logn)|²

n .

This is the case becauseµ=P

n≥1 δ±logn

n is a sampling measure forPW.

Bohr correspondence

Another very important source of information onH² is the Bohr correspondence. This is the identification ofH² with H²(T^∞).

Definition

We endowT^∞ with the Haar measureµwhich is the product measure. We denote byH²(T^∞) the subspace generated by z^α =z₁^α1· · ·z_k^αk where α are all finite multiindex with positive entries. i.e. α1 ≥0, . . . , αk ≥0.

The functions onH²(T^∞) represent holomorphic functions of infinitely many variables. To anyf ∈H²(T^∞) we associate a power seriesf(z) =P

αa_αz^α, whereP

|a_α|² <∞. This series is not convergent (in general) whenz ∈D^∞. We need that

P|z|^2α<+∞ and this is the case exactly whenz ∈`2 because X

α

|z|^2α=Y

j

(1− |z_j|²)⁻¹.

Thusf ∈H²(T^∞) represent holomorphic functions in D^∞∩`₂.

Infinitely many variables

To be precise, what is a function inH²(D^∞)? Given a formal series of infinitely many variablesf(z) =P

a_αz^α, and givenm∈N, we definef_m :D^m→Cas the power series where we replace

(z₁, . . . ,z_m,z_m+1, . . .)→(z₁, . . . ,z_m,0,0, . . .).

Thenf ∈H²(D^∞) if and only iffm∈H²(D^m) with uniform control of the norms.

kf_mk²

H²(D^m) = sup

r<1

Z

T^m

|f(rz)|².

Whenp= 2 this is not really needed but this definition works for anyp∈[1,∞]

Bohr correspondence

The correspondence that Bohr established is the following: To any functionf ∈ H² we consider the functionF ∈H²(T^∞) via the following relationship:

f(s) =X

n

ann^−s ←→F(z) =X

α

bαz^α, wherebα =an when n=p^α₁¹· · ·p^α_k^k is the prime number decomposition ofn.There is an alternative way of looking at this identification: Consider the one complex dimensional curveγ in D^∞,γ :C⁺_1/2 →D^∞ defined asγ(s) = (p^−s₁ ,p₂^−s, . . .). Clearly for alls ∈C⁺_1/2,γ(s)∈`₂∩D^∞ and f ∈ H if and only if

f(s) =F(γ(s)) for some (unique) F ∈H²(D^∞).

Vertical convolution

A good smoothing approximation to functions onH² and related spaces that are invariant under translations are vertical

convolutions. Letφ∈L¹(R), φ≥0 and with integral one. Let f be a Dirichlet series. Then ifs ≥σB we may define

f_ε(s) = Z ∞

−∞

f(s +it)φ(^t_ε)

ε dt.

One checks that iff(s) =P

a_nn^−s, then f_ε(s) =

∞

X

n=1

a_nφ(εˆ logn)n^−s

We will use asφ(t) = _2π^{1 sin}_x^x2

. This ensures thatf_ε is always a Dirichlet polynomial. Moreoverfε→f in H² and uniformly in σ > σ_B+δ.

Multipliers

This is for instance a useful way to describe the pointwise multipliers inH². We say that g is a multiplier inH² whenever gH² ⊂ H². In particular, since 1∈ H² then any multiplier belongs toH².Lindqvist, Hedenmalm and Seip described the multipliers.

For this we need the following definition:

Definition

We say that f ∈ H^∞ if f ∈ H² and moreover f extends analytically to a bounded function inC⁺₀.

Theorem

The multipliers ofH² are exactlyH^∞and the multiplier norm of g coincides withsup

C⁺0 |g|.

Multipliers

This can be seen because the functions inH^∞ are in

correspondence with functions inH^∞(T^∞). These are functions f ∈L^∞(T^∞) such thatR

T^∞f(z)¯z^αdµ(z) = 0 for all positive multiindices. These functions define bounded holomorphic functions inD^∞∩c0.

Thus we are actually transferring the result that the multiplier of H²(T^∞) is H^∞(T^∞) and functions in H^∞(T^∞) are holomorphic in the bigger setD^∞∩c0 and the curve γ(C⁺₀)⊂D^∞∩c0.

Multipliers

More precisely ifg is a multiplier we may assume (taking vertical convolutions) thatg is a Dirichlet polynomial and we want to proof thatkgk_H^∞ ≤ kgk_M.

Now we may identifyg with an holomorphic polynomial ind variables that is a multiplier inH²(T^∞), in particular inH²(T^d).

By iteration of the multiplier we know that kg^k1k_L2(T^d) ≤ kgk^k_Mk1k_L2(T^d)

Thus Z

T^d

|g|^2k ≤ kgk^2k_M.

LetAε⊂T^d be the set where |g| ≥(1 +ε)kgkM, then µ(A_ε)^1/2k(1 +ε)kgk_M ≤ kgk_M and if follows thatkgk_L∞(T^d)≤ kgk_M.

Riesz basis of dilates

One application: Letφ∈L²(0,1) and consider it extended as an odd periodic function of period 2. Beurling asked for whichφ the system of dilates ofφ

φ(x), φ(2x), φ(3x), . . .

is a Riesz basis inL²(0,1). The prototypical example is

φ(x) = sin(πx) when we have an orthonormal basis.For a generalφ we express it as

φ(x) =X

n≥1

a_nsin(nπx).

The solution is given in terms ofa_n. More precisely, let f(s) =P

n≥1ann^−s ∈ H².

Theorem (Hedenmalm, Lindqvist, Seip)

The system of dilates ofφis a Riesz basis for L²(0,1)if and only if f ∈ H^∞ and0< ε <|f(s)|for all s ∈C⁺₀.

Riesz basis of dilates

The proof can be seen as follows. Define two operators: T,S as T :L²(0,1)→L²(0,1); T(sin(nπx)) =φ(nx)

S :L²(0,1)→ H²; S(sin(nπx)) =n^−s.

ClearlyS is an invertible operator since it sends an orthonormal basis to another. On the other handφ(nx) is a Riesz basis if and only ifT is invertible.

Let us check that for any Dirichlet polynomialp(s) STS⁻¹p =f(s)p(s)

Takep(s) =n^−s, then

STS⁻¹(p) =S(φ(nx)) =S(X

k

aksin(knπx)) = Xak(kn)^−s =f(s)n^−s =f(s)p(s).

Vertical limits

We have seen that the multipliers “live” in a bigger half-planeC⁺₀. In fact, in a sense, many functions inH² are defined there.

Given a functionf(s) =P

nann^−s ∈ H² if we consider its vertical translateg(s) =f(s+it) it has coefficients:

g(s) =X

n

a_ne^−itlognn^−s.

We can generalize it. Consider a multiplicative character χ:N→S¹,χ(nm) =χ(n)χ(m) and|χ(n)|= 1. Then given f ∈ H² we can define

f_χ(s) =X

n

χ(n)a_nn^−s.

The vertical translates are particular cases, but they are dense.

That is for any givenχ there is a sequence of vertical translates such that

f_χ= lim

n f(s+it_n).

uniformly on compacts ofC⁺_1/2

Vertical translates

Why this is so?

This is again a consequence of Kronecker theorem. Any character χcan be identified with a point in T^∞. The characters are determined by its values on the primesχ(p1), χ(p2), . . . and these can be chosen independently, each one inT.

So every character is given by the point inT^∞: χ(p₁), χ(p₂), . . .

The vertical translates correspond to the characters χt(pj) =e^itlogpj

and these are dense inT^∞.

Vertical translates

In the space of characters we can consider the probability measure mwhich is the Haar measure in T^∞(the product measure). Then Helson proved:

Theorem (Helson)

Given f ∈ H², for almost all characters χthe vertical limit function fχ extends toC^1/2₀ .

Moreover there is pointwise convergence in the imaginary axis:

Theorem (Hedenmalm, Saksman)

Given f ∈ H², for almost all charactersχ and almost all t ∈R, the seriesP

a_nχ(n)n^−it is convergent.

and we can compute the norm off through the vertical translates:

Theorem (Hedenmalm, Lindqvist, Seip)

Letµbe an absolutely continuous probability measure on R. kfk²₂=

Z

T^∞

Z

R

|f_χ(it)|²dµ(t)dm(χ).

A curiosity on the Riemann hypothesis

This has the following curious corollary.

Proposition

If a_n is totally multiplicative and in`₂ then f_χ(s) =P

a_nχ(n)n^−s converges for almost all characters to a zero free function inC⁺₀. The proof: The functionfχ(s) converges to the half planeC⁺₀ as mentioned and moreover

1/fχ(s) =X

n

µ(n)anχ(n)n^−s

whereµ(n) is the Mobious function. Thus 1/f_χ also extends to C⁺₀, therefore f_χ is zero free.

Corollary (Helson)

Almost all vertical translates of the Riemannζ function ζχ(s) =X

χ(n)n^−s, are zero free inC⁺_1/2.

L

variants

Let 1≤p ≤+∞. We can try to extend the theory of Dirichlet series inH² toH^p as proposed by Bayart. There are two possible equivalent definitions:

Definition

H^p is the closure of Dirichlet polynomials d(s)with the norm kdk^p_p:= lim

T

1 T

Z T 0

|d(it)|^pdt

or alternatively Definition

f ∈ H^p if f(s) =P

ann^−s and there is a function ˜f ∈L^p(T^∞) such thatR

T^∞

˜fz¯^αdµ(z) =a_n(α) for all multiindicesα with all entries positive andR

T^∞

˜f¯z^αdµ(z) = 0 otherwise.

L

variants

The Bohr correspondence works well but the theory inH^p is much less developed. One of the most basic pieces of information lacking is the local embedding, i.e. is it true that

Z 1 0

|f(1/2 +it)|^pdt ≤Ckfk^p.

for any Dirichlet polynomialf?The answer is true ifp = 2n but we cannot interpolate between the spacesH^p.

The casep = 1 will follow if any functionf ∈ H¹ weakly factorizes asf =P

jgjhj, with gj,hj ∈ H² andkfk₁'P

jkg_jk₂kh_jk₂. This is known to be true for functions inH¹(Tⁿ).

Unfortunately this is not the case inH¹(T^∞).

Theorem

There are functions inH¹ that cannot be weakly-factorized.

The Bourgain-Montgomery conjecture

The conjecture is not too unsimilar to a conjecture formulated by Montgomery and refined by Bourgain. Letf(s) =P

n≤Na_nn^−s be a Dirichlet polynomial of degreeN with coefficients |a_n| ≤1. Then

Z T 0

|f(it)|^qdt ≤C(T +N^q/2)N^q/2+ε.

This is known ifq = 2n (it is the version of the local embedding that we proved!) but it is open for other values ofq.

Composition operators (boundedness)

Gordon and Hedenmalm have studied the composition operators in H². More precisely they studied the following question:

Question

For which analytic mappingsφ:C⁺_1/2 →C⁺_1/2 is the composition operator C_φ(f) =f ◦φa bounded linear operator on H²? Their complete answer is the following:

Theorem

C_φ is a bounded composition operator if and only if:

I φ(s) =c0s +ψ(s), where c0 ∈N∪ {0} andψ(s)is an ordinary Dirichlet series converging in some point.

I φadmits an analytic extension to C⁺₀ such that

I φ(C⁺0)⊂C⁺0 if c0>0, and

I φ(C⁺0)⊂C⁺_1/2if c₀= 0.

Composition operators (compactness)

Several authors (Bayart, Finet, Lef`evre, Queff´elec, Volberg, at least) have studied the compactness of the composition operators inH². So far, the description is not complete. We give one of the results in this area:

Theorem (Bayart)

Letφ:C⁺₀ →C⁺₀ be of the formφ(s) =c₀s+ψ(s) c₀ ≥1.

Suppose

I =ψ is bounded onC⁺₀

I N_φ(s) =o(σ)as σ →0.

Then C_φis compact on H². In this statement

N_φ(s) = (P

w∈φ⁻¹(s)<w if s ∈φ(C⁺₀)

0 otherwise.

Different weights

McCarthy introduced analogous spaces of Dirichlet series that behave locally like the Bergman space and obtained similar properties. This theme has been further studied by Olsen. His setting is the following:

Definition

LetH²_ω be the space of ordinary Dirichlet series such that f(s) =X

n≥1

a_nn^−s : X|a_n|²

ω_n <+∞.

The main results studied are the local behaviour of functions in the spaceH²_ω in terms of the weight. Suppose that the weightω satisfies

A x

(logx)^α ≤X

n≤x

ω_n≤B x

(logx)^α

for someα ≤1.Then the main result is that under this hypothesis on the weightH_ω² behave locally as functions inD_α(C⁺_1/2).

Different weights

The functions inDα(C⁺_1/2) correspond to weighted Bergman spaces forα <0:

Z

C⁺_1/2

|f(s)|²(σ−1/2)^−1−αdm(s)<+∞.

Whenα= 0,D_α(C⁺_1/2) is the Hardy spaceH²(C⁺_1/2).

When 0< α≤1, they are weighted Dirichlet spaces:

Z

C⁺_1/2

|f⁰(s)|²(σ−1/2)^+1−αdm(s)<+∞.

By local behaviour we mean the local embedding, the description of compact Carleson measures, the bounded interpolation

sequences and local extension as in the Hardy-Dirichlet space.

Estimates for the coefficients P

|a

|

LetD_N be a Dirichlet polynomial, then We have

N

X

n=1

|a_n| ≤







(1 +o(1))√

NlogNkD_Nk₁

√NkD_Nk₂

√ Ne^−(1/

√2+o(1))√

logNlog logNkD_Nk_∞,

Estimates for the coefficients P

|a

|

The first estimate on terms ofkD_Nk₁ follows from the isoperimetric type inequality due to Bayart and Helson:

N

X

n=1

|a_n|²

d(n) ≤ kD_Nk₁ whered(n) is the number of divisor of n.

We may estimate then by Cauchy-Schwartz and using some known estimates forPN

n=1d(n) we get

N

X

n=1

|a_n| ≤(1 +o(1))p

NlogNkD_Nk₁

The estimate in terms of kD

k

We letS(N) be the smallest constantC such that the inequality P

n=1|a_n| ≤CkD_Nk_∞ holds for every nontrivialD_N.

Theorem (Konyagin–Queff´elec 2001, de la Bret`eche 2008, DFOOS 2011)

We have S(N) =

√ Nexpn

− 1

√2 +o(1)p

logNloglogNo when N → ∞.

A refined version of Bohr theorem

With the estimates forS(N) and the estimate sup

σ>0

X

n≤N

ann^−s

≤C logNsup

σ>0

X

n≥1

ann^−s one can prove that:

Theorem

The supremum of the set of real numbers c such that

∞

X

n=1

|a_n|n⁻¹²exp n

cp

lognlog logn o

<∞

for everyP∞

n=1ann^−s in H^∞ equals 1/√ 2.

This is really a refinement of a theorem of R. Balasubramanian, B. Calado, and H. Queff´elec. The basic point is the conection of Dirichlet series with power series in the infinite polydisk.

Sidon sets

Definition

IfG is an Abelian compact group and Γ its dual group, a subset of the charactersS ⊂Γ is called a Sidon set if

X

γ∈S

|α_γ| ≤CkX

γ∈S

α_γγk_∞

The smallest constantC(S) is called the Sidon constant ofS. We estimate the Sidon constant for homogeneous polynomials Definition

ThenS(m,n) is the smallest constantC such that the inequality P

|α|=m|a_α| ≤CkPk_∞ holds for everym-homogeneous polynomial inn complex variablesP(z) =P

|α|=ma_αz^α.

The Sidon constant for homogeneous polynomials

Proposition

Let m and n be positive integers larger than1. Then S(m,n)≤e√

m(√ 2)^m−1

n+m−1 m

^m−1_2m .

Note that we also have the following trivial estimate:

S(m,n)≤ s

n+m−1 m

,

This inequality is essentially sharp. This is a corollary of the hypercontractivity of the Bohnenblust and Hille estimate for polynomials.

A multilinear inequality

In 1930, Littlewood proved that for every bilinear form B:Cⁿ×Cⁿ→Cwe have

X

i,j

|B(eⁱ,e^j)|^4/3 3/4

≤√ 2 sup

z,w∈Dⁿ

|B(z,w)|,

This was extended by Bohnenblust and Hille who proved in 1931:

X

i1,...,im

|B(eⁱ¹, . . . ,e^im)|^{m+12m m+1}_2m

≤√

2^m−1 sup

zⁱ∈Dⁿ

|B(z¹, . . . ,z^m)|.

The exponent _m+1^2m cannot be improved if one wants constants independent of the number of variables.

A symmetric Bohnenblust-Hille inequality

Our main result is that the polynomial Bohnenblust–Hille inequality is in fact hypercontractive:

Theorem

Let m and n be positive integers larger than1. Then we have X

|α|=m

|a_α|^{m+12m m+1}_2m

≤e√ m(

√

2)^m−1 sup

z∈Dⁿ

X

|α|=m

aαz^α

for every m-homogeneous polynomialP

|α|=ma_αz^α onCⁿ.

Polarization

There is an obvious relationship between both identities.If we restrict a symmetric multilinear form to the diagonal

P(z) =B(z, . . . ,z), then we obtain a homogeneous polynomial.

The converse is also true: Given a homogeneous polynomial P :Cⁿ→Cof degreem, by polarization, we may define the symmetric m-multilinear formB :Cⁿ× · · · ×Cⁿ→C so that B(z, . . . ,z) =P(z).

B(z¹, . . . ,z^m) = 1 2^mm!

X

εi=±1 1≤i≤m

ε₁ε₂· · ·ε_mPX^m

i=1

ε_izⁱ

Harri’s lemma states that

|B(z¹, . . . ,z^m)| ≤ m^m m!kPk_∞.

Two lemmas

Lemma (Blei)

For all sequences(c_i)_i where i = (i₁, . . . ,i_m) and i_k = 1, . . . ,n, we have

Xⁿ

i1,...,im=1

|c_i|^{m+12m m+1}_2m

≤ Y

1≤k≤m

hXⁿ

ik=1

X

i1,...,ik−1,ik+1,...,im

|c_i|²¹₂i_m¹ .

Lemma (Bayart)

For any homogeneous polynomial P(z) = P

|α|=m

aαz^α onCⁿ:

X

|α|=m

|a_α|²¹

2 ≤(√ 2)^m

X

|α|=m

aαz^α L¹(Tⁿ).

The proof 1/3

We denote the polynomialp as p(z) = X

i1≤···≤im

c(i1, . . . ,im)zi1· · ·zim. We have

X

i1≤···≤im

|c(i1, . . . ,im)|^2m/(m+1) ≤ X

i1,...,im

|c(i1, . . . ,im)|

|i|^1/2

2m/(m+1)

If we use now Blei’s lemma, the last sum is bounded by

m

Y

k=1

hX^m

i_k=1

X

i^k

|c(i1, . . . ,im)|²

|i|

1/2i1/m

≤√ m

m

Y

k=1

hX^m

i_k=1

X

i^k

|i^k||c(i1, . . . ,im)|²

|i|²

1/2i1/m

.

The proof 2/3

Now observe that we freeze the variablei_k and we group the terms to make a polynomial again:

X

i^k

|i^k||c(i₁, . . . ,i_m)|²

|i|²

1/2

=X

i^k

|i^k||B(eⁱ¹, . . . ,e^im)|²1/2

=kP_kk₂. wherePk(z) is the polynomial

P_k(z) =B(z, . . . ,z,e^ik,z, . . . ,z).Now we use Bayart estimate and X

i^k

|i^k||c(i1, . . . ,im)|²

|i|²

1/2

≤√ 2^m−1

Z

Tⁿ

|B(z, . . . ,z,e^ik,z, . . . ,z)|.

The proof 3/3

We replacee^ik byλe^ik with |λ|= 1. We take τ_k(z) =P

λ_k(z)e^ik in such a way that

n

X

ik=1

X

i^k

|i^k||c(i₁, . . . ,im)|

|i|²

1/2

≤ Z

Tⁿ

B(z, . . . , τ_k(z), . . . ,z)

≤e√

2^m−1kPk∞. Finally we have

X

i1≤···≤im

|c(i₁, . . . ,im)|^2m/(m+1)≤e

√ 2^m−1√

mkPk_∞.

Open Problems

I Describe the bounded zero sequences of functions in H².

I Prove the “baby corona” ^problem, i.e. if f₁ andf₂ are multipliers in H², such that its associated functions F1,F2 ∈H^∞(D^∞) satisfy|F₁(z)|+|F₂(z)| ≥δ for all z ∈c0∩D^∞ andg is an arbitrary function inH², are there functions g₁,g₂ ∈ H² such thatf₁g₁+f₂g₂ =g?

I Disprove the embedding problem forH¹,i.e. prove that the (local) Carleson measures of H¹ are different from the (local) Carleson measures ofH².

I Describe the compact composition operators on H².

Bibliography *used* to elaborate this notes

R. Balasubramanian, B. Calado, and H. Queff´elec.

The Bohr inequality for ordinary Dirichlet series.

Studia Math., 175(3):285–304, 2006.

F. Bayart, S. V. Konyagin, and H. Queff´elec.

Convergence almost everywhere and divergence everywhere of Taylor and Dirichlet series.

Real Anal. Exchange, 29(2):557–586, 2003/04.

Fr´ed´eric Bayart.

Hardy spaces of Dirichlet series and their composition operators.

Monatsh. Math., 136(3):203–236, 2002.

Fr´ed´eric Bayart.

Compact composition operators on a Hilbert space of Dirichlet series.

Illinois J. Math., 47(3):725–743, 2003.

Fr´ed´eric Bayart, Catherine Finet, Daniel Li, and Herv´e Queff´elec.

Composition operators on the Wiener-Dirichlet algebra.

J. Operator Theory, 60(1):45–70, 2008.

Harold P. Boas.

The football player and the infinite series.

Notices Amer. Math. Soc., 44(11):1430–1435, 1997.

H. F. Bohnenblust and Einar Hille.

On the absolute convergence of Dirichlet series.

Ann. of Math. (2), 32(3):600–622, 1931.

Fritz Carlson.

Contributions `a la th´eorie des s´eries de Dirichlet. IV.

Ark. Mat., 2:293–298, 1952.

Andreas Defant, Leonhard Frerick, Joaquim Ortega-Cerd`a, Myriam Ouna¨ıes, and Kristian Seip.

The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive.

Ann. of Math. (2), 174(1):485–497, 2011.

Catherine Finet, Herv´e Queff´elec, and Alexander Volberg.

Compactness of composition operators on a Hilbert space of Dirichlet series.

J. Funct. Anal., 211(2):271–287, 2004.

Julia Gordon and H˚akan Hedenmalm.

The composition operators on the space of Dirichlet series with square summable coefficients.

Michigan Math. J., 46(2):313–329, 1999.

H˚akan Hedenmalm.

Dirichlet series and functional analysis.

In the Legacy of Niels Henrik Abel, The Abel Bicentennial, Oslo 2002 (O. A. Laudal, R. Piene, editors), Springer-Verlag, pages 673–684, 2004.

H˚akan Hedenmalm, Peter Lindqvist, and Kristian Seip.

A Hilbert space of Dirichlet series and systems of dilated functions inL²(0,1).

Duke Math. J., 86(1):1–37, 1997.

H˚akan Hedenmalm, Peter Lindqvist, and Kristian Seip.

Addendum to: “A Hilbert space of Dirichlet series and systems of dilated functions inL²(0,1)” [Duke Math. J. 86(1997), no.

1, 1–37; MR1427844 (99i:42033)].

Duke Math. J., 99(1):175–178, 1999.

H˚akan Hedenmalm and Eero Saksman.

Carleson’s convergence theorem for Dirichlet series.

Pacific J. Math., 208(1):85–109, 2003.

Henry Helson.

Foundations of the theory of Dirichlet series.

Acta Math., 118:61–77, 1967.

Henry Helson.

Dirichlet series.

Henry Helson, Berkeley, CA, 2005.

S. V. Konyagin and H. Queff´elec.

The translation ¹₂ in the theory of Dirichlet series.

Real Anal. Exchange, 27(1):155–175, 2001/02.

Brian Maurizi and Herv´e Queff´elec.

Some remarks on the algebra of bounded Dirichlet series.

J. Fourier Anal. Appl., 16(5):676–692, 2010.

John E. McCarthy.

Hilbert spaces of Dirichlet series and their multipliers.

Trans. Amer. Math. Soc., 356(3):881–893 (electronic), 2004.

Jan-Fredrik Olsen.

Modified zeta functions as kernels of integral operators.

J. Funct. Anal., 259(2):359–383, 2010.

Jan-Fredrik Olsen.

Local properties of hilbert spaces of dirichlet series.

To appear in Journal Functional Analysis, 2011.

Jan-Fredrik Olsen and Eero Saksman.

On the boundary behaviour of the hardy spaces of dirichlet series and a frame bound estimate.

To appear in J. Reine Angew. Math., 2009.

Jan-Fredrik Olsen and Kristian Seip.

Local interpolation in Hilbert spaces of Dirichlet series.

Proc. Amer. Math. Soc., 136(1):203–212 (electronic), 2008.

H. Queff´elec.

H. Bohr’s vision of ordinary Dirichlet series; old and new results.

J. Anal., 3:43–60, 1995.

Eero Saksman.

An introduction to hardy spaces of dirichlet series.

http://www.crm.cat/Activitats/Activitats/2010- 2011/wkspectral/Abstracts/Saksman.pdf, 2011.

Eero Saksman and Kristian Seip.

Integral means and boundary limits of Dirichlet series.

Bull. Lond. Math. Soc., 41(3):411–422, 2009.

Caution!

The true corona problem, i.e. inH^∞ can be missleading. This is incorrectly formulated:

Wrong problem

Considerf1,f2 ∈ H^∞ such that|f₁|+|f₂| ≥δ >0 in C⁺₀. Is it true that there are functionsg1,g2 ∈ H^∞ such thatf1g1+f2g2 = 1?

This is not right. Takef1(s) = 2^−s −1/5 and f2(s) = 3^−s −1/5.

If there were solutionsg₁,g₂∈ H^∞, then there will bounded holomorphic functionsG₁,G₂∈H^∞(D²) such that

(z1−1/5)G1(z1,z2) + (z2−1/5)G2(z1,z2) = 1!

Right problem (too hard)

Considerf₁,f₂ ∈ H^∞ such that its associated functions

F1,F2∈ H^∞(D^∞) satisfy|F₁|+|F₂| ≥δ >0 inc0∩D^∞. Is it true that there are functionsg₁,g₂ ∈ H^∞ such thatf₁g₁+f₂g₂ = 1?

This seems very hard because one needs to solve the Corona theorem in the polydisk.