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A Survey of Recent Results on the Hardy Space of Dirichlet Series

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A Survey of Recent Results on the Hardy Space of Dirichlet Series A Survey of Recent Results on the Hardy Space 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-Hilbert space on D, a Hilbert space of holomorphic functions on D with a square summable power series. ( 1 1 ) 2 X n X 2 H (D) = f(z) = anz : janj < 1 n=0 n=0 where the inner product is given by * 1 1 + 1 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 1 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 norm is equivalent to a growth condition on the radial boundary values of its functions, so P1 n that if f(z) = n=0 anz , 1 X Z 2π dt kfk2 = ja j2 = sup jf(reit)j2 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 2 H (D) to functions f~ 2 H (T) ⊆ L (T), where Z 2π dt Z 2π dt kfk2 = sup jf(reit)j2 = jf~(eit)j2 = 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 < 1, we can form the Hardy space Hp similarly, with Z 2π dt Z 2π dt kfkp = sup jf(reit)jp = jf~(eit)jp = 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 Lp space 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 fzng 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. P1 n 2 If λ 2 D and f(z) = n=0 anz 2 H (D), then the reproducing P1 n n kernel kλ(z) = n=0 λ z is such that 1 1 X n X n f(λ) = anλ = an(λ ) = hf; kλiH2(D) n=0 n=0 P1 n 2 2 Note that n=0 λ < 1, so that kλ 2 H (D). Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Zero Sets of H2(D) Given a sequence of points fzng ⊆ D, there is a nontrivial function 2 f 2 H (D) which vanishes at each zn if and only if fzng satisfies the Blaschke condition 1 X (1 − jznj) < 1 n=1 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville Multiplier Algebra of H2(D) 2 1 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 1 p p H (C+) = f 2 Hol(C+) : sup jf(σ + it)j dt < 1 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 2 C+ and f 2 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 1 dt < 1, so that k 2 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 fzng ⊆ C+, there is a nontrivial 2 function f 2 H (C+) which vanishes at each zn if and only if fzng satisfies the following condition 1 X xn < 1 1 + jz j2 n=1 n where zn = xn + iyn. If the sequence fzng is bounded, then we recover a Blaschke-type condition 1 X xn < 1 n=1 Survey of Hardy-Dirichlet Series Spaces University of Tennessee, Knoxville 2 Multiplier Algebra of H (C+) 2 1 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 P1 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 P1 an For a particular f(s) = n=1 ns , we write ( 1 ) X an σ (f) = inf R(s): converges c ns n=1 ( 1 ) X an σ (f) = inf ρ : converges to a bounded function in Ω b ns ρ n=1 ( 1 ) X an σ (f) = inf ρ : converges uniformly in Ω u ns ρ n=1 ( 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 p The Riesz-Fischer theorem states that '(x) = 2 sin(πx) can be dilated to form a complete orthonormal basis np p o 2 sin(πx); 2 sin(π2x);::: = f'(x);'(2x);:::g 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 ' can take the place of sin so that f'(nx)gn≥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 1 X p '(x) = an 2 sin(nπx) n=1 into 1 X an S'(s) = ns n=1 and analyzing properties of the analytic S'. 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 f'(nx)gn≥1 is a Reisz basis for L (0; 1) if and only if S' and 1=S' 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-Dirichlet space), ( 1 1 ) X an X H2 = f(s) = : ja j2 < 1 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 1 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 1 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 2 H have σa ≤ 2 , as by Cauchy-Schwarz 1 !2 1 1 X an X 2 X 1 ≤ janj ns n2s n=1 n=1 n=1 This bound is sharp, since 1 1 p X 1 X n= log(n + 1) = 2 H2 ns−1=2 log(n + 1) ns n=1 n=1 2 so H ⊂ Hol(C1=2).
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