[Paper Review] Characterizations of product Hardy spaces in Bessel setting

Xuan Thinh Duong, Ji Li+2 more|arXiv (Cornell University)|Jun 10, 2016

Advanced Harmonic Analysis Research55 references3 citations

TL;DR

This paper establishes equivalent characterizations of product Hardy spaces associated with Bessel operators using Bessel Riesz transforms, non-tangential and radial maximal functions via Poisson and heat semigroups. It extends classical multiparameter harmonic analysis beyond the Laplacian by proving atomic decompositions and duality for product BMO spaces in the Bessel setting, marking a significant advancement over the Chang–Fefferman framework.

ABSTRACT

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt--Stein, especially the generalised Cauchy--Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the extension of Merryfield's result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators. These results are a first extension for product Hardy and BMO associated to a differential operator other than the Laplacian and are a major step beyond the Chang--Fefferman setting.

Motivation & Objective

• To extend multiparameter harmonic analysis to Bessel operators, moving beyond the classical Laplacian framework.
• To establish equivalent characterizations of product Hardy spaces in the Bessel setting using maximal functions and Riesz transforms.
• To develop atomic decompositions and duality for product BMO spaces associated with Bessel operators.
• To generalize Merryfield’s result connecting non-tangential maximal functions and area functions to the Bessel setting.
• To apply the grand maximal function technique to link non-tangential and radial maximal functions in product Bessel spaces.

Proposed method

• Utilizes the Bessel Laplace equation and generalized Cauchy–Riemann equations from Muckenhoupt–Stein theory to define harmonic functions in the Bessel setting.
• Applies atomic decomposition techniques adapted to the Bessel operator framework to characterize product Hardy spaces.
• Employs Poisson and heat semigroups to define non-tangential and radial maximal functions for product spaces.
• Uses the grand maximal function technique to connect non-tangential and radial maximal functions, enabling duality results.
• Applies Littlewood–Paley theory and area function methods to relate maximal functions and Riesz transforms in the Bessel setting.
• Leverages the $T1$ theorem and Coifman–Weiss space of homogeneous type framework to extend product Hardy and BMO theory to non-Laplacian operators.

Experimental results

Research questions

• RQ1 How can product Hardy spaces be characterized in the Bessel setting using maximal functions and Riesz transforms?
• RQ2 What is the relationship between non-tangential maximal functions and area functions in the product Bessel setting?
• RQ3 Can the grand maximal function technique be extended to link radial and non-tangential maximal functions in product Bessel spaces?
• RQ4 How do atomic decompositions of product Hardy spaces in the Bessel setting compare to classical ones in the Laplacian case?
• RQ5 What is the duality structure of the product BMO space associated with Bessel operators?

Key findings

• The paper establishes equivalent characterizations of product Hardy spaces in the Bessel setting via Bessel Riesz transforms, non-tangential and radial maximal functions.
• It proves that the non-tangential maximal function is equivalent to the area function in the Bessel product setting, generalizing Merryfield’s result.
• The grand maximal function technique successfully links non-tangential and radial maximal functions, enabling the derivation of product BMO duality.
• The decomposition of the product BMO space associated with Bessel operators is obtained directly through maximal function techniques.
• The $L^p$ boundedness of the grand maximal function operator is established, with uniform bounds independent of parameters, ensuring the finiteness of the $L^p$ norm of the maximal function.
• The proof shows that $\|u^*\|_{L^p(\mathbb{R}_\lambda)}^p \lesssim 1$, confirming the boundedness of the maximal function in the product Bessel setting.