On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.

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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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On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

Cited by 18 publications

References 13 publications

The multiplicative Hilbert matrix

The multiplicative Hilbert matrix

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

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

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