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Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series

Daniel Carando, Andreas Defant, Felipe Marceca, Ingo Schoolmann, Pablo Sevilla-Peris

Abstract

For a general Dirichlet series $\sum a_n e^{-λ_n s}$ with frequency $λ=(λ_n)_n$, we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips $S_p(λ)$ that encode the minimum translation necessary for series in the Hardy space $\mathcal{H}_p(λ)$ to have absolutely convergent coefficients.

Hypercontractivity and strips of convergence in Hardy spaces of general Dirichlet series

Abstract

For a general Dirichlet series with frequency , we study how horizontal translation (i.e. convolution with a Poisson kernel) improves its integrability properties. We characterize hypercontractive frequencies in terms of their additive structure answering some questions posed by Bayart. We also provide sharp bounds for the strips that encode the minimum translation necessary for series in the Hardy space to have absolutely convergent coefficients.
Paper Structure (8 sections, 18 theorems, 136 equations)

This paper contains 8 sections, 18 theorems, 136 equations.

Table of Contents

  1. Introduction and main results
  2. The examples
  3. The proofs
  4. Preliminaries
  5. Localization
  6. Hypercontractivity
  7. Strips
  8. Optimality

Key Result

Theorem 1.1

Let $\lambda$ be a frequency. The following are equivalent: If additionally $\lambda$ satisfies that $\limsup \tfrac{\log\log n}{\lambda_n}=0$, both are also equivalent to:

Theorems & Definitions (39)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Remark 1.4
  • Proposition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Proposition 1.9
  • Proposition 1.10
  • ...and 29 more