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

Bondarenko, A. A.; Brevig, Ole Fredrik; Saksman, Eero; Seip, Kristian

doi:10.1090/tran/7898

Cited by 16 publications

(20 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem and Lemma have applications in the theory of Hardy spaces of Dirichlet series, as will be exhibited in the forthcoming paper .…”

Section: Hardy–littlewood Inequalitiesmentioning

confidence: 98%

“…Only the bound for

q ⩾ 2

in the following theorem will be used in the proof of Theorem . Since the proofs are similar, and both bounds are of intrinsic interest, we have found it natural to treat the whole range

0 < q < \infty

; the bound for

q ⩽ 2

will find applications in . Theorem If

0true f (s) = \sum_{n = 1}^{N} a_{n} n^{- s}

, then

{((), \sum_{n = 1}^{N} \frac{| a_{n} {false|}^{2}}{Φ_{2 / q} (n)})}^{\frac{1}{2}} ⩽ {∥ f ∥}_{q}, q ⩽ 2,

{∥ f ∥}_{q} ⩽ {((), \sum_{n = 1}^{N} {false| a_{n} false|}^{2} Φ_{q / 2} (n))}^{\frac{1}{2}}, q ⩾ 2 .

…”

Section: Hardy–littlewood Inequalitiesmentioning

confidence: 99%

“…Only the bound for q 2 in the following theorem will be used in the proof of Theorem 1. Since the proofs are similar, and both bounds are of intrinsic interest, we have found it natural to treat the whole range 0 < q < ∞; the bound for q 2 will find applications in [4].…”

Section: Hardy-littlewood Inequalitiesmentioning

confidence: 99%

“…From a functional analytic point of view, the underlying operator is that of partial sums acting on Hardy spaces of Dirichlet series:

0true S_{N} ((), \sum_{n = 1}^{\infty} a_{n} n^{- s}) : = \sum_{n = 1}^{N} a_{n} n^{- s} .

Following Bayart , we define the Hardy space

H^{q}

for

0 < q < \infty

by taking the closure of all Dirichlet polynomials

f

with respect to the norm (or quasi‐norm when

0 < q < 1

)

{∥ f ∥}_{q} : = {((), lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {false| f (i t) false|}^{q} d t)}^{1 / q} .

The aforementioned theorem of Helson shows that when

1 < q < \infty

, there is a uniform bound on the norm of

S_{N}

when it acts on

H^{q}

. We refer to the forthcoming paper for some new estimates on the growth of the norm of

S_{N}

in the interesting range

0 < q ⩽ 1

. The computation of the pseudomoments deals with the special situation when

S_{N}

acts on functions with a strong multiplicative structure.…”

Section: Introductionmentioning

confidence: 99%

“…The aforementioned theorem of Helson [18] shows that when 1 < q < ∞, there is a uniform bound on the norm of S N when it acts on H q . We refer to the forthcoming paper [4] for some new estimates on the growth of the norm of S N in the interesting range 0 < q 1. The computation of the pseudomoments deals with the special situation when S N acts on functions with a strong multiplicative structure.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Pseudomoments of the Riemann zeta function

Bondarenko

Brevig

Saksman

et al. 2018

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

The 2kth pseudomoments of the Riemann zeta function ζ(s) are, following Conrey and Gamburd, the 2kth integral moments of the partial sums of ζ(s) on the critical line. For fixed k>1/2, these moments are known to grow like false(prefixlogNfalse)k2, where N is the length of the partial sum, but the true order of magnitude remains unknown when k⩽1/2. We deduce new Hardy–Littlewood inequalities and apply one of them to improve on an earlier asymptotic estimate when k→∞. In the case k<1/2, we consider pseudomoments of ζαfalse(sfalse) for α>1 and the question of whether the lower bound false(prefixlogNfalse)k2α2 known from earlier work yields the true growth rate. Using ideas from recent work of Harper, Nikeghbali and Radziwiłł and some probabilistic estimates of Harper, we obtain the somewhat unexpected result that these pseudomements are bounded below by logN to a power larger than k2α2 when k<1/e and N is sufficiently large.

show abstract

“…Theorem and Lemma have applications in the theory of Hardy spaces of Dirichlet series, as will be exhibited in the forthcoming paper .…”

Section: Hardy–littlewood Inequalitiesmentioning

confidence: 98%

“…Only the bound for

q ⩾ 2

in the following theorem will be used in the proof of Theorem . Since the proofs are similar, and both bounds are of intrinsic interest, we have found it natural to treat the whole range

0 < q < \infty

; the bound for

q ⩽ 2

will find applications in . Theorem If

0true f (s) = \sum_{n = 1}^{N} a_{n} n^{- s}

, then

{((), \sum_{n = 1}^{N} \frac{| a_{n} {false|}^{2}}{Φ_{2 / q} (n)})}^{\frac{1}{2}} ⩽ {∥ f ∥}_{q}, q ⩽ 2,

{∥ f ∥}_{q} ⩽ {((), \sum_{n = 1}^{N} {false| a_{n} false|}^{2} Φ_{q / 2} (n))}^{\frac{1}{2}}, q ⩾ 2 .

…”

Section: Hardy–littlewood Inequalitiesmentioning

confidence: 99%

Section: Hardy-littlewood Inequalitiesmentioning

confidence: 99%

“…From a functional analytic point of view, the underlying operator is that of partial sums acting on Hardy spaces of Dirichlet series:

0true S_{N} ((), \sum_{n = 1}^{\infty} a_{n} n^{- s}) : = \sum_{n = 1}^{N} a_{n} n^{- s} .

Following Bayart , we define the Hardy space

H^{q}

for

0 < q < \infty

by taking the closure of all Dirichlet polynomials

f

with respect to the norm (or quasi‐norm when

0 < q < 1

)

{∥ f ∥}_{q} : = {((), lim_{T \to \infty} \frac{1}{T} \int_{0}^{T} {false| f (i t) false|}^{q} d t)}^{1 / q} .

The aforementioned theorem of Helson shows that when

1 < q < \infty

, there is a uniform bound on the norm of

S_{N}

when it acts on

H^{q}

. We refer to the forthcoming paper for some new estimates on the growth of the norm of

S_{N}

in the interesting range

0 < q ⩽ 1

. The computation of the pseudomoments deals with the special situation when

S_{N}

acts on functions with a strong multiplicative structure.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Pseudomoments of the Riemann zeta function

Bondarenko

Brevig

Saksman

et al. 2018

Bull. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces

2021

Self Cite

View full text Add to dashboard Cite

We describe the idempotent Fourier multipliers that act contractively on $$H^p$$ H p spaces of the d-dimensional torus $$\mathbb {T}^d$$ T d for $$d\ge 1$$ d ≥ 1 and $$1\le p \le \infty $$ 1 ≤ p ≤ ∞ . When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $$L^p$$ L p spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $$p=2(n+1)$$ p = 2 ( n + 1 ) , with n a positive integer, contractivity depends in an interesting geometric way on n, d, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $$H^p(\mathbb {T}^\infty )$$ H p ( T ∞ ) for every $$1 \le p \le \infty $$ 1 ≤ p ≤ ∞ and that extends to a bounded operator if and only if $$p=2,4,\ldots ,2(n+1)$$ p = 2 , 4 , … , 2 ( n + 1 ) .

show abstract

Convergence and almost sure properties in Hardy spaces of Dirichlet series

Bayart

2021

Math. Ann.

View full text Add to dashboard Cite

Given a frequency λ, we study general Dirichlet series ane −λns . First, we give a new condition on λ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in the right half-plane converges uniformly in this half-plane, improving classical results of Bohr and Landau. Then, following recent works of Defant and Schoolmann, we investigate Hardy spaces of these Dirichlet series. We get general results on almost sure convergence which have an harmonic analysis flavour. Nevertheless, we also exhibit examples showing that it seems hard to get general results on these spaces as spaces of holomorphic functions.

show abstract

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

Cited by 16 publications

References 34 publications

Pseudomoments of the Riemann zeta function

Pseudomoments of the Riemann zeta function

Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces

Convergence and almost sure properties in Hardy spaces of Dirichlet series

Contact Info

Product

Resources

About