A Survey of Recent Results on the Hardy of Dirichlet Series

Gregory Zitelli

University of Tennessee, Knoxville

September 2013

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Notation

D is the open unit disc. T is the unit circle. Cρ is the right half plane with real part > ρ. C+ = C0.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

We begin with the standard definition of the Hardy- on D, a Hilbert space of holomorphic functions on D with a square summable power series.

( ∞ ∞ ) 2 X n X 2 H (D) = f(z) = anz : |an| < ∞ n=0 n=0

where the inner product is given by

* ∞ ∞ + ∞ X n X n X hf, giH2(D) = anz , bnz = anbn n=0 n=0 H2(D) n=0

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

2 This formulation of the Hardy-Hilbert space H (D) is useful 2 because it emphasizes the canonical equivalence of H (D) and 2 ` (N), namely

∞ X n f(z) = anz ∼ (a0, a1, a2,...) n=0

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

Interestingly, the Hardy-Hilbert space is equivalent to a growth condition on the radial boundary values of its functions, so P∞ n that if f(z) = n=0 anz , ∞ X Z 2π dt kfk2 = |a |2 = sup |f(reit)|2 H2(D) n 2π n=0 0≤r<1 0

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

Hardy-Hilbert space functions living on D have nontangential boundary values almost everywhere on T, allowing us to extend 2 2 2 functions f ∈ H (D) to functions f˜ ∈ H (T) ⊆ L (T), where

Z 2π dt Z 2π dt kfk2 = sup |f(reit)|2 = |f˜(eit)|2 = kf˜k2 H2(D) H2(T) 0≤r<1 0 2π 0 2π

2 2 and H (T) is the subspace of L (T) whose elements have only nonnegative Fourier coefficients.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

For general 1 < p < ∞, we can form the Hardy space Hp similarly, with

Z 2π dt Z 2π dt kfkp = sup |f(reit)|p = |f˜(eit)|p = kf˜kp Hp(D) Hp(T) 0≤r<1 0 2π 0 2π

p ∼ p p p p Here H (D) = H (T) ⊆ L (T). We treat H as both H (D) and p H (T) interchangeably.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

The Hardy spaces Hp can be thought of both as the holomorphic functions on D which satisfy a growth condition on the boundary, and the nontangential boundary functions which live inside of the on that boundary.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces

There are three important properties posessed by the Hardy space H2 as a Hilbert space which we would like to contrast:

Reproducing kernels kλ

Zero sets {zn} Multiplier algebra M(H2)

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Reproducing Kernels for the space H2(D)

2 Point evaluations are bounded linear functionals on H (D), and can therefore be expressed as inner products with appropriate reproducing kernels.

P∞ n 2 If λ ∈ D and f(z) = n=0 anz ∈ H (D), then the reproducing P∞ n n kernel kλ(z) = n=0 λ z is such that ∞ ∞ X n X n f(λ) = anλ = an(λ ) = hf, kλiH2(D) n=0 n=0

P∞ n 2 2 Note that n=0 λ < ∞, so that kλ ∈ H (D).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Zero Sets of H2(D)

Given a of points {zn} ⊆ D, there is a nontrivial function 2 f ∈ H (D) which vanishes at each zn if and only if {zn} satisfies the Blaschke condition

∞ X (1 − |zn|) < ∞ n=1

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Multiplier Algebra of H2(D)

2 ∞ The multipliers M(H (D)) are precisely H (D), the bounded holomorphic functions on the disc.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Spaces in Half Planes

There is similarly a Hardy space for the half plane C+ using the growth condition on the imaginary line

 Z ∞  p p H (C+) = f ∈ Hol(C+) : sup |f(σ + it)| dt < ∞ σ>0 −∞

p One can also define spaces H (Cρ) for arbitrary ρ. Like the Hardy 2 2 space H (D), H (C+) has well understood reproducing kernels, zero sets, and a multiplier algebra.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville 2 Reproducing Kernels for the space H (C+)

2 Point evaluations are bounded linear functionals on H (C+), and can therefore be expressed as inner products with appropriate reproducing kernels.

2 If λ ∈ C+ and f ∈ H (C+), then the reproducing kernel k (z) = 1 is such that λ z+λ

f(λ) = hf, kλiH2(D)

2 Note that sup R ∞ 1 dt < ∞, so that k ∈ H2( ). x>0 −∞ x+it+λ λ D

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville 2 Zero Sets of H (C+)

Given a sequence of points {zn} ⊆ C+, there is a nontrivial 2 function f ∈ H (C+) which vanishes at each zn if and only if {zn} satisfies the following condition

∞ X xn < ∞ 1 + |z |2 n=1 n

where zn = xn + iyn. If the sequence {zn} is bounded, then we recover a Blaschke-type condition

∞ X xn < ∞ n=1

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville 2 Multiplier Algebra of H (C+)

2 ∞ The multipliers M(H (C+)) are precisely H (C+), the bounded holomorphic functions on the right half plane.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Dirichlet Series

P∞ an A Dirichlet series is a series of the form f(s) = n=1 ns . We write s = σ + it, and let Ωρ denote the half plane with real part > ρ.

Unlike power series, the “radius” of convergence and absolute convergence may be different.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Dirichlet Series

P∞ an For a particular f(s) = n=1 ns , we write

( ∞ ) X an σ (f) = inf R(s): converges c ns n=1 ( ∞ ) X an σ (f) = inf ρ : converges to a bounded function in Ω b ns ρ n=1 ( ∞ ) X an σ (f) = inf ρ : converges uniformly in Ω u ns ρ n=1 ( ∞ ) X an σ (f) = inf R(s): converges absolutely a ns n=1

σc ≤ σb = σu ≤ σa ≤ σc + 1

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Riesz-Basis

√ The Riesz-Fischer theorem states that ϕ(x) = 2 sin(πx) can be dilated to form a complete orthonormal basis n√ √ o 2 sin(πx), 2 sin(π2x),... = {ϕ(x), ϕ(2x),...}

for L2(0, 1).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Riesz-Basis

A natural extension of the theorem would be to ask which functions ϕ can take the place of sin so that {ϕ(nx)}n≥1 forms an orthonormal basis for L2(0, 1) under an equivalent norm. Such a sequence is called a Riesz basis.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Riesz-Basis

The characterization of Riesz-type sets which are complete in L2(0, 1) was characterized by Beurling in 1945, by transforming the expression

∞ X √ ϕ(x) = an 2 sin(nπx) n=1 into

∞ X an Sϕ(s) = ns n=1 and analyzing properties of the analytic Sϕ. In 1995, Hedenmalm, Lindqvist, and Seip solved the Reisz-basis problem completely by exploiting a Hilbert space of analytic functions of this form. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Riesz-Basis

Theorem (Hedenmalm, Lindqvist, Seip) 2 The system {ϕ(nx)}n≥1 is a Reisz basis for L (0, 1) if and only if Sϕ and 1/Sϕ are in the multiplier algebra M(H2).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Space of Dirichlet Series

The proof used the Hardy space of Dirichlet series (or Hardy-),

( ∞ ∞ ) X an X H2 = f(s) = : |a |2 < ∞ ns n n=1 n=1

along with the characterization of the multipliers M(H2) of the space. The paper also further established Bohr’s work on the connection between the Hardy-Dirichlet space H2 and the Hardy 2 ∞ space of the infinite polycircle H (T ).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Space of Dirichlet Series

The work by Hedehmalm, Lindqvist, and Seip inspired an investigation of the space H2 and various related spaces over the next 15 years. Contributors in analysis include Aleman, Andersson, Bayart, McCarthy, Olsen, Saskman.

Topics included Multipliers Reproducing kernels Zero sets for H2 and related Hp spaces Boundary behavior (What happens on the line σ = 1/2? Can you look at behavior of the function on the line σ = 0?) p ∞ Connections with the infinite polycircle H (T ) Carleson measures

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Space of Dirichlet Series

The condition on the Hardy-Dirichlet space ensures that all 2 1 functions f ∈ H have σa ≤ 2 , as by Cauchy-Schwarz ∞ !2 ∞ ∞ X an X 2 X 1 ≤ |an| ns n2s n=1 n=1 n=1

This bound is sharp, since

∞ ∞ √ X 1 X n/ log(n + 1) = ∈ H2 ns−1/2 log(n + 1) ns n=1 n=1

2 so H ⊂ Hol(C1/2).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hardy Space of Dirichlet Series

The Hardy-Dirichlet space H2 clearly mirrors the classical Hardy space H2 of the disc

( ∞ ∞ ) X an X H2 = f(s) = : |a |2 < ∞ ns n n=1 n=1 ( ∞ ∞ ) 2 X n X 2 H = f(z) = anz : |an| < ∞ n=0 n=0

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Reproducing Kernels in H2

The reproducing kernels on H2 are actually quite easy to define, P∞ an 2 since if f(s) = n=1 ns ∈ H and λ ∈ C1/2 then

∞ ∞ ∞ X an X X f(λ) = = a e−λ log(n) = a e−λ log(n) nλ n n n=1 n=0 n=0 ∞ * ∞ + X  1  X 1 = an = f(s), nλ ns+λ n=0 n=1 H2

so that kλ(s) = ζ(s + λ).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Zero Sets of H2

2 Like the space H (C+), bounded {zn} have the same Blaschke-type condition that

∞ X (xn − 1/2) < ∞ n=1

On the other hand, Dirichlet series have strange vertical limit behavior which has made it difficult to fully classify unbounded sequences.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Almost Periodic Behavior of Dirichlet Series

If a function f ∈ Hol(Cρ),  > 0, then we say that t is an -translation number for f if

sup |f(s + it) − f(s)| ≤  s∈Cρ

We say that f is uniformly almost periodic if for every  > 0 there is a length M such that every interval of length M contains at least one -translation number for f.

Theorem If f ∈ Hol(Cρ) is represented by a Dirichlet series which converges uniformly in Cρ, then f is uniformly almost periodic.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Zero Sets of H2

2 Since all functions in H are uniformly almost periodic in C1/2, they will either have no zeros or infinitely many zeros, and those zeros may be distributed quite wildly along vertical strips.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Multiplier Algebra of H2

The multiplier algebra of H2 consist precisely of those holomorphic functions in C+ which are bounded and representable by a Dirichlet series. If D is used to denote the collection of holomorphic functions representable by a convergence Dirichlet series on some half space, then we can write

2 ∞ 2 M(H ) = H (C+) ∩ D = M(H (C+)) ∩ D

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville “Arms-Reach” Boundary Condition

Theorem (Carlson’s Lemma)

P∞ an If f(s) = n=1 ns is convergent and bounded in C+, then ∞ Z T 2 X 2 1 2 kfkH2 = |an| = lim lim |f(σ + it)| dt σ→0+ T →∞ 2T n=1 −T

Note that σ moves all the way back to C+ rather than just C1/2, and requires that f be convergent and bounded in C+ to begin with. An interesting extension of H2 is as follows.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hp Spaces

For 1 ≤ p < ∞ we define the space Hp as the closure of the Dirichlet polynomials under the norm

p 1/p Z ∞ ∞ ! 1 X an lim dt T →∞ 2T nit −∞ n=1

We will refer to this as the Hp norm. Bayart (2002) showed that the Dirichlet series which represent such functions have σu ≤ 1/2, so that they represent holomorphic functions on C1/2.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Notes on D∞ and T∞

k Let T be the k-dimensional polycircle, and let p1, . . . , pk it it enumerate the first k primes. Then the injection (p1 , . . . , pk ) for k t ∈ R has dense range in T .

∞ This means that there is a dense subset of the infinite polydisc D −s −s such that its elements can be expressed as z = (p1 , p2 ,...), where s = σ + it such that t ∈ R and σ > 0.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hp Spaces as Hp(T∞)

α1 α2 αr Let each n factor into n = p1 p2 ··· pr . Setting −s −s z = (p1 , p2 ,...) we have

∞ ∞ X an X f(s) = = a zα1 ··· zαr ns n 1 r n=1 n=1

P∞ α1 αr We consider the transformation Df(z) = n=1 anz1 ··· zr as ∞ being a function of z ∈ D .

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Hp Spaces as Hp(T∞) Bohr showed that in fact for Dirichlet polynomials PN an P (s) = n=1 ns ,

p Z ∞ N p 1 X an kP k p = lim dt H T →∞ 2T nit −∞ n=1 p 1 Z ∞ N X −it −it = lim anp1 ··· p dt T →∞ 2T k −∞ n=1 p Z N X α1 pk = anz1 ··· zk dm(z) = kDP kHp(T∞) ∞ T n=1

Which follows from properties of the Kronecker flow and Birkhoff-Khintchin theorem. Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville p Proof that H ⊆ Hol(C1/2)

−ω −ω ∞ Let ω ∈ C1/2, and z = (2 , 3 ,...) ∈ D . Note that since 1 2 <(ω) > 2 , z ∈ ` as well.

p p ∞ p ∞ If f ∈ H , then Df ∈ H (T ). On H (T ) we have the inequality

kDfkp ∞ p p Hp(T∞) p Y 1 |f(ω)| = |Df(z)| ≤ = kfk p Q∞ (1 − |z |2) H 1 − |p−2ω| j=1 j j=1 j

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville p Proof that H ⊆ Hol(C1/2)

kDfkp ∞ p p Hp(T∞) p Y 1 |f(ω)| = |Df(z)| ≤ = kfk p Q∞ (1 − |z |2) H 1 − |p−2ω| j=1 j j=1 j

However, Euler’s identity concerning the Riemann zeta function says that the last term is precisely

∞ X 1 kfkp Hp n2<(ω) n=1

which is simply finite by p-series. Consequently,

p p |f(ω)| ≤ kfkHp ζ(2<(ω))

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Almost-Sure Properties of Hp

∞ An element χ ∈ T can be thought of as a character in the sense that it acts on the prime elements in the canonical way. We define p the function fχ where f ∈ H is the function to be influence by the character as

∞ X anχ(n) f (s) = χ ns n=1

Theorem p ∞ For f ∈ H and for almost every χ ∈ T , fχ is a Dirichlet series which converges in C+.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville p p Comparing H to H (C1/2)

Since Hp are spaces of Dirichlet series which are well defined in p C1/2, it is interesting to note comparisons between H and p H (C1/2). Theorem (Hedenmalm, Lindqvist, Seip) 2 2 If f ∈ H then f(s)/s ∈ H (C1/2).

In particular, this tells us that all functions in H2 have nontangential boundary values (as do functions in Hp).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville p p Comparing H to H (C1/2)

p p Let H∞(C1/2) denote the uniform local H space of the right half plane, defined as those elements such that

Z y+1 p p kfk p = sup sup |f(σ + it)| dt < ∞ H∞(C1/2) y∈R σ>1/2 y

Theorem (Bayart) p p If p ≥ 2, then H ⊂ H∞(C1/2) and the injection is continuous.

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Carleson Measures

Theorem (Bayart)

If 1 ≤ p < ∞ and µ is a positive measure on C1/2, and if µ is a Carleson for Hp then it is also a for p H (C1/2).

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Open Problems

p p Does H embed in H∞(C1/2) for 1 ≤ p < 2? Is there a BMOA theory for the Hp spaces? 2 2 Can H be factored like H (D)? What kind of classification for zero-sets can be achieved in the Hp setting?

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Thank you!

Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville