[Paper Review] Integration operators between Hardy spaces on the unit ball of $\Cn$

Jordi Pau|arXiv (Cornell University)|Dec 3, 2013

Advanced Harmonic Analysis Research32 references3 citations

TL;DR

This paper provides a complete characterization of the boundedness and compactness of integration operators $J_g$ acting between Hardy spaces $H^p(\mathbb{B}_n)$ and $H^q(\mathbb{B}_n)$ on the unit ball in $\mathbb{C}^n$, for all $0 < p, q < \infty$. It establishes that $J_g$ is bounded from $H^p$ to $H^q$ if and only if $g$ belongs to specific function spaces—$BMOA$ for $p=q$, Lipschitz-type spaces $\Lambda(\alpha)$ with $\alpha = n(1/p - 1/q)$ for $p < q$, and $H^r$ with $1/r = 1/q - 1/p$ for $p > q$, with sharp norm estimates provided.

ABSTRACT

We completely describe the boundedness of the Volterra type operator $J_ g$ between Hardy spaces in the unit ball of $\Cn$. The proof of the one dimensional case used tools, such as the strong factorization for Hardy spaces, that are not available in higher dimensions, and therefore new techniques are developed. In particular, a generalized version of the description of Hardy spaces in terms of the area function is needed.

Motivation & Objective

• To close a significant gap in the literature by fully characterizing the boundedness of the Volterra-type integration operator $J_g$ between Hardy spaces $H^p(\mathbb{B}_n)$ and $H^q(\mathbb{B}_n)$ for all $0 < p, q < \infty$.
• To extend one-dimensional results of Aleman-Siskakis and Aleman-Cima to the higher-dimensional setting of the unit ball in $\mathbb{C}^n$, where classical tools like strong factorization for Hardy spaces are unavailable.
• To develop new techniques tailored to higher dimensions, particularly a generalized characterization of Hardy spaces via the area function, and to establish sharp norm estimates for $J_g$ in terms of function space norms of $g$.
• To address the non-diagonal cases ($p \ne q$) and the critical case $p = q$, providing a complete and unified framework for the operator's behavior across all $p, q$.

Proposed method

• The proof relies on a generalized characterization of Hardy spaces in terms of the area function, which is essential for extending one-dimensional results to $\mathbb{C}^n$.
• A key technical tool is the use of tent spaces and Carleson measures, particularly through the identification of the dual of $H^p$ via tent space duality.
• The author employs dyadic decomposition of the sphere $\mathbb{S}_n$ into Whitney-type cubes $Q$, and defines associated tent regions $\widehat{Q}$, to localize the analysis and control the operator norm.
• The norm of $J_g$ is estimated via testing against sequences $\lambda_Q$ supported on dyadic cubes, leading to a comparison with the $L^r$ norm of a maximal function $\widetilde{\mu}$ associated with the measure $\mu$.
• Khinchine’s inequality is applied to control $L^p$-norms of random sums over dyadic cubes, enabling the derivation of $\ell^p$-type estimates for the sequence $\lambda_Q$.
• A Vitali-type covering lemma is used to relate the measure of tent regions $\widehat{Q}$ to the surface measure of their bases, allowing the transfer of $L^r$-norm estimates on the sphere to the measure $\mu$.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on $g \in H(\mathbb{B}_n)$ for the integration operator $J_g$ to be bounded from $H^p(\mathbb{B}_n)$ to $H^q(\mathbb{B}_n)$ for all $0 < p, q < \infty$?
• RQ2 How does the boundedness of $J_g$ depend on the smoothness or integrability properties of $g$ in higher dimensions, particularly when classical tools like strong factorization fail?
• RQ3 What is the precise relationship between the operator norm $\|J_g\|_{H^p \to H^q}$ and the norm of $g$ in relevant function spaces such as $BMOA$, $\Lambda(\alpha)$, or $H^r$?
• RQ4 Can the boundedness of $J_g$ be characterized using Carleson measures and tent space duality in the higher-dimensional setting, and if so, how?
• RQ5 What role does the radial derivative $Rg$ play in determining the boundedness of $J_g$, and how is this reflected in the function space norms of $g$?

Key findings

• For $p = q$, $J_g$ is bounded on $H^p$ if and only if $g \in BMOA$, with $\|J_g\| \asymp \|g\|_{BMOA}$, providing a sharp norm estimate.
• For $p < q$, $J_g: H^p \to H^q$ is bounded if and only if $g \in \Lambda(\alpha)$ with $\alpha = n(1/p - 1/q)$, and $\|J_g\|_{H^p \to H^q} \asymp \|g\|_{\Lambda(\alpha)}$, with the condition that $\alpha \leq 1$; if $\alpha > 1$, then $g$ must be constant.
• For $p > q$, $J_g: H^p \to H^q$ is bounded if and only if $g \in H^r$ with $1/r = 1/q - 1/p$, and $\|g\|_{H^r} \asymp \|J_g\|_{H^p \to H^q}$, establishing a precise duality between the $H^r$-norm of $g$ and the operator norm.
• The proof introduces a generalized area function characterization of Hardy spaces in $\mathbb{C}^n$, which is essential for overcoming the absence of strong factorization in higher dimensions.
• The paper establishes a new duality framework using tent spaces and Carleson measures, enabling the derivation of sharp norm estimates via dyadic decomposition and covering lemmas.
• The results are sharp in the sense that the function space conditions on $g$ are both necessary and sufficient, and the norm comparisons are quantitative and optimal.