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Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series

Carlos Gómez-Cabello, Pascal Lefèvre, Hervé Queffélec

Abstract

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional.

Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series

Abstract

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional.
Paper Structure (8 sections, 12 theorems, 65 equations)

This paper contains 8 sections, 12 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Dirichlet series and the spaces Hp and A
  3. Dirichlet series
  4. The Hp spaces
  5. The A-spaces
  6. The Dirichlet-Riemann-Liouville operator
  7. A word on the integration operator
  8. Main result

Key Result

Lemma 2.1

Let $p\geq1$, $\alpha>-1$ and $s\in{\mathbb C}_{\frac{1}{2}}$. We have

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • ...and 17 more