Local properties of Hilbert spaces of Dirichlet series

Olsen, Jan-Fredrik

doi:10.1016/j.jfa.2011.07.007

Cited by 17 publications

(39 citation statements)

References 31 publications

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“…The connection between the operator Z K,I and the frame (3.1) was essentially observed in the paper [23]. There the operator Z N,I was used to study the Dirichlet-Hardy space H 2 = n∈N a n n −s : n∈N |a n | 2 < +∞ .…”

Section: Discussionmentioning

confidence: 99%

“…This gives meaning to the notation F (1/2 + it) for F ∈ H 2 . A special case of the main result of [23] is now stated as follows.…”

Section: Discussionmentioning

confidence: 99%

“…We make some remarks on related work. The operator Z N,I appears in the context of Hilbert spaces of Dirichlet series in a paper by J.-F. Olsen and E. Saksman [23] (see also Section 7). In work done by J.-P. Kahane [12], a closely related functional is used to give a proof of the classical prime number theorem.…”

Section: ) Impliesmentioning

confidence: 99%

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Modified zeta functions as kernels of integral operators

Olsen

2010

Journal of Functional Analysis

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The modified zeta functions n∈K n −s , where K ⊂ N, converge absolutely for Re s > 1. These generalise the Riemann zeta function which is known to have a meromorphic continuation to all of C with a single pole at s = 1. Our main result is a characterisation of the modified zeta functions that have pole-like behaviour at this point. This behaviour is defined by considering the modified zeta functions as kernels of certain integral operators on the spaces L 2 (I ) for symmetric and bounded intervals I ⊂ R. We also consider the special case when the set K ⊂ N is assumed to have arithmetic structure. In particular, we look at local L p integrability properties of the modified zeta functions on the abscissa Re s = 1 for p ∈ [1, ∞].

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Section: Discussionmentioning

confidence: 99%

“…This gives meaning to the notation F (1/2 + it) for F ∈ H 2 . A special case of the main result of [23] is now stated as follows.…”

Section: Discussionmentioning

confidence: 99%

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Modified zeta functions as kernels of integral operators

Olsen

2010

Journal of Functional Analysis

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“…Here it is crucial that Φ α is strictly positive. By (25) and [24,Thm. 1] it follows that there is some C α such that…”

Section: Proof Of Theorem 31 (B)mentioning

confidence: 99%

Linear space properties of $H^p$ spaces of Dirichlet series

Bondarenko¹,

Brevig²,

Saksman³

et al. 2019

Trans. Amer. Math. Soc.

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We study H p spaces of Dirichlet series, called H p , for the range 0 < p < ∞. We begin by showing that two natural ways to define H p coincide. We then proceed to study some linear space properties of H p . More specifically, we study linear functionals generated by fractional primitives of the Riemann zeta function; our estimates rely on certain Hardy-Littlewood inequalities and display an interesting phenomenon, called contractive symmetry between H p and H 4/p , contrasting the usual L p duality. We next deduce general coefficient estimates, based on an interplay between the multiplicative structure of H p and certain new one variable bounds. Finally, we deduce general estimates for the norm of the partial sum operator ∞ n=1 a n n −s → N n=1 a n n −s on H p with 0 < p ≤ 1, supplementing a classical result of Helson for the range 1 < p < ∞. The results for the coefficient estimates and for the partial sum operator exhibit the traditional schism between the ranges 1 ≤ p ≤ ∞ and 0 < p < 1.

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“…The case when j = 1 was established in [9]. We shall use the same method to establish the general case.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

Zhao

2018

Concrete Operators

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The Hilbert spaces H w consisiting of Dirichlet series F (s) = ∞ n=1 a n n −s that satisfty ∞ n=1 |a n | 2 /w n < ∞, with {w n } n of average order log j n (the j -fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon-Hedenmalm theorem on such H w from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ(s) = c 0 s + φ(s), where c 0 is a nonnegative integer and φ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c 0 = 0. It is verified for every integer j 1, real α > 0 and {w n } n having average order (log + j n) α , that the composition operators map H w into a scale of H w ′ with w ′ n having average order (log + j +1 n) α . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.

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Local properties of Hilbert spaces of Dirichlet series

Cited by 17 publications

References 31 publications

Modified zeta functions as kernels of integral operators

Modified zeta functions as kernels of integral operators

Linear space properties of $H^p$ spaces of Dirichlet series

Iteration of Composition Operators on small Bergman spaces of Dirichlet series

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