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Characterization of multipliers on vector-valued Hardy spaces

Jorge Antezana, Daniel Carando, Tomás Fernández Vidal, Melisa Scotti

TL;DR

The paper advances the theory of pointwise multipliers for vector-valued Hardy spaces in infinite-variable settings, showing that multipliers on the infinite polydisk align with $H_ ty( ext{D}^ abla_2,B(X))$ while those on the infinite polytorus require the weaker $H_ ty^{sot}( ext{T}^ abla_ ty,B(X))$ when $X$ is separable. It further establishes a precise bridge between these two multiplier spaces via the $P_ ty$ mapping, which becomes an isomorphism under the Analytic Radon-Nikodym property. Extending to vector-valued Dirichlet series, the authors characterize multipliers for $ ext{H}_p^+(X)$ and $ ext{H}_p(X)$ through $ ext{H}_ ty^+(B(X))$ and $ ext{H}_ ty^{sot}(B(X))$, respectively, with ARNP yielding a unified picture. Overall, the work clarifies how domain geometry and Banach-space properties govern multiplier behavior in infinite-dimensional Hardy spaces and their Dirichlet-series analogues.

Abstract

This work characterizes the multipliers on vector-valued Hardy spaces over the infinite polydisk and the infinite polytorus, as well as in the context of Dirichlet series. Unlike the scalar-valued setting, where these frameworks are completely analogous reformulations of one another, there are significant differences in the vector-valued context. We prove that while the space of multipliers on the infinite polydisk is $H_\infty(\mathbb{D}^\infty_2, B(X))$, the situation on the infinite polytorus is distinct; assuming $X$ is separable, the multiplier space can be identified as $H_\infty^{sot}(\mathbb{T}^\infty, B(X))$, consisting of essentially bounded SOT-measurable functions. These spaces coincide when $X$ possesses the analytic Radon-Nikodym property. Finally, we extend these results to the associated Hardy spaces of Dirichlet series, $\mathcal{H}_p^+(X)$ and $\mathcal{H}_p(X)$, providing characterizations for their respective multiplier spaces.

Characterization of multipliers on vector-valued Hardy spaces

TL;DR

The paper advances the theory of pointwise multipliers for vector-valued Hardy spaces in infinite-variable settings, showing that multipliers on the infinite polydisk align with while those on the infinite polytorus require the weaker when is separable. It further establishes a precise bridge between these two multiplier spaces via the mapping, which becomes an isomorphism under the Analytic Radon-Nikodym property. Extending to vector-valued Dirichlet series, the authors characterize multipliers for and through and , respectively, with ARNP yielding a unified picture. Overall, the work clarifies how domain geometry and Banach-space properties govern multiplier behavior in infinite-dimensional Hardy spaces and their Dirichlet-series analogues.

Abstract

This work characterizes the multipliers on vector-valued Hardy spaces over the infinite polydisk and the infinite polytorus, as well as in the context of Dirichlet series. Unlike the scalar-valued setting, where these frameworks are completely analogous reformulations of one another, there are significant differences in the vector-valued context. We prove that while the space of multipliers on the infinite polydisk is , the situation on the infinite polytorus is distinct; assuming is separable, the multiplier space can be identified as , consisting of essentially bounded SOT-measurable functions. These spaces coincide when possesses the analytic Radon-Nikodym property. Finally, we extend these results to the associated Hardy spaces of Dirichlet series, and , providing characterizations for their respective multiplier spaces.

Paper Structure

This paper contains 10 sections, 15 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Hardy spaces
  3. Holomorphic functions on the polydisk
  4. Hardy spaces on the polytorus
  5. The connection between the polydisk and the polytorus
  6. Multipliers of vector-valued Hardy spaces
  7. On the polydisk
  8. On the polytorus
  9. The connection between multiplier spaces.
  10. Multipliers in vector-valued Dirichlet series

Key Result

Proposition 2.1

DirSer The $X$-valued analytic trigonometric polynomials are norm-dense in $H_p(\mathbb{T}^\infty, X)$ for every $1 \leq p < \infty$, and $w\left(L_\infty(\mathbb{T}^\infty, X), L_1(\mathbb{T}^\infty, X^*)\right)$-dense in $H_\infty(\mathbb{T}^\infty, X)$.

Theorems & Definitions (34)

  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 24 more