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Factorization of Hardy-Orlicz Space on the Disk and applications to Hankel Operators

Jean-Marcel Tanoh Dje, Justin Feuto

TL;DR

This work extends classical Hardy space factorization to Hardy–Orlicz spaces on the disk, establishing an inner–outer decomposition for $G \in H^{\Phi}(\mathbb{D})$ with $\|G\|_{H^{\Phi}}^{lux} \asymp \|O_G\|_{H^{\Phi}}^{lux}$ and proving a strong, multiplicative factorization when $\Phi_3^{-1} \sim \Phi_1^{-1} \Phi_2^{-1}$ so that any $G \in H^{\Phi_3}(\mathbb{D})$ splits as $G=G_1 G_2$ with $G_j \in H^{\Phi_j}(\mathbb{D})$ and $\|G\|_{H^{\Phi_3}}^{lux} \asymp \|G_1\|_{H^{\Phi_1}}^{lux}\|G_2\|_{H^{\Phi_2}}^{lux}$. These factorization results are then employed to characterize the boundedness of Hankel operators between Hardy–Orlicz spaces: in one regime, $h_b$ is bounded from $H^{\Phi_1}$ to $H^{\Phi_2}$ iff $b \in H^{\Phi_3}$ with $\Phi_3^{-1} \sim \Phi_1^{-1} \Phi_2^{-1}$ (norm equivalence $\|h_b\| \asymp \|b\|_{H^{\Phi_3}}^{lux}$); in another regime, a $BMOA(\varrho)$-type condition on $b$ provides a gain estimate when $\Phi_1,\Phi_2$ satisfy suitable convexity/duality assumptions. The paper thus unifies and extends classical Riesz and Volberg–Tolokonnikov factorization to Hardy–Orlicz spaces and derives sharp Hankel-symbol criteria in this broader setting, with duality links to $BMOA(\varrho)$ spaces.

Abstract

In this work, we prove that the product of a function belonging to a Hardy-Orlicz space $H^{Φ_{1}}$ and a function from another Hardy-Orlicz space $H^{Φ_{2}}$ belongs to a third Hardy-Orlicz space $H^{Φ_{3}}$. Moreover, we establish the converse: any holomorphic function in the space $H^{Φ_{3}}$ can be expressed as the product of two functions, one from $H^{Φ_{1}}$ and the other from $H^{Φ_{2}}$. Subsequently, we use this factorization result in Hardy-Orlicz spaces to study the continuity of the Hankel operator in these spaces. More specifically, we provide gain and loss estimates for the norms of the Hankel operator in the context of analyzing its continuity in Hardy-Orlicz spaces.

Factorization of Hardy-Orlicz Space on the Disk and applications to Hankel Operators

TL;DR

This work extends classical Hardy space factorization to Hardy–Orlicz spaces on the disk, establishing an inner–outer decomposition for with and proving a strong, multiplicative factorization when so that any splits as with and . These factorization results are then employed to characterize the boundedness of Hankel operators between Hardy–Orlicz spaces: in one regime, is bounded from to iff with (norm equivalence ); in another regime, a -type condition on provides a gain estimate when satisfy suitable convexity/duality assumptions. The paper thus unifies and extends classical Riesz and Volberg–Tolokonnikov factorization to Hardy–Orlicz spaces and derives sharp Hankel-symbol criteria in this broader setting, with duality links to spaces.

Abstract

In this work, we prove that the product of a function belonging to a Hardy-Orlicz space and a function from another Hardy-Orlicz space belongs to a third Hardy-Orlicz space . Moreover, we establish the converse: any holomorphic function in the space can be expressed as the product of two functions, one from and the other from . Subsequently, we use this factorization result in Hardy-Orlicz spaces to study the continuity of the Hankel operator in these spaces. More specifically, we provide gain and loss estimates for the norms of the Hankel operator in the context of analyzing its continuity in Hardy-Orlicz spaces.

Paper Structure

This paper contains 11 sections, 25 theorems, 111 equations.

Table of Contents

  1. Introduction and statement of main results.
  2. Some definitions and useful properties.
  3. Growth functions.
  4. Hardy-Orlicz Space on $\mathbb{D}$.
  5. Proof of the main results.
  6. Factorization of Hardy-Orlicz Spaces.
  7. Proof of Theorem \ref{['pro:main2a']}.
  8. Proof of Theorem \ref{['pro:main0apaaal0']}.
  9. Proof of Theorem \ref{['pro:main2aqop']}.
  10. Applications to Hankel Operators.
  11. Proofs of Theorem \ref{['pro:mainfqmaqppaaqqmaqq5']} and Theorem \ref{['pro:madgaaqqmaqq5']}.

Key Result

Theorem 1.1

Let $\Phi$ be a growth function of lower type. Let $0\not\equiv G\in H^{\Phi}(\mathbb{D})$ and $B$ the Blaschke product associated with the zeros sequence of $G$. The function $G/B$ belongs to $H^{\Phi}(\mathbb{D })$ and $\|G\|_{H^{\Phi}}^{lux}\approx\left\|G/B\right\|_{H^{\Phi}}^{lux}$.

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 27 more