[Paper Review] Characterizations of $H^1_{Δ_N}(\mathbb{R}^n)$ and $ m BMO_{Δ_N}(\mathbb{R}^n)$ via Weak Factorizations and Commutators

Ji Li, Brett D. Wick|arXiv (Cornell University)|May 17, 2015

Advanced Harmonic Analysis Research12 references3 citations

TL;DR

This paper establishes equivalent characterizations of the Hardy space $H^1_{\Delta_N}(\mathbb{R}^n)$ and BMO space $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ associated with the Neumann Laplacian $\Delta_N$, using weak factorizations and commutators with Riesz transforms. It proves that $H^1_{\Delta_N}$ admits equivalent norms via maximal functions, Riesz transforms, atoms, and weak factorizations, while $\text{BMO}_{\Delta_N}$ is characterized by the boundedness of commutators with Riesz transforms and by the action of these transforms on $L^\infty$ functions.

ABSTRACT

This paper provides a deeper study of the Hardy and $ m BMO$ spaces associated to the Neumann Laplacian $Δ_N$. For the Hardy space $H^1_{Δ_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space $H^1(\mathbb{R}^n)$) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space ${ m BMO}_{Δ_N}(\mathbb{R}^n)$ (which contains the classical $ m BMO(\mathbb{R}^n)$) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neumann Laplacian on $L^\infty(\mathbb{R}^n)$ functions and in terms of the behavior of the commutator with the Riesz transforms. The results obtained extend many of the fundamental results known for $H^1(\mathbb{R}^n)$ and $ m BMO(\mathbb{R}^n)$.

Motivation & Objective

• To extend classical characterizations of $H^1(\mathbb{R}^n)$ and $\text{BMO}(\mathbb{R}^n)$ to the setting of the Neumann Laplacian $\Delta_N$.
• To establish equivalent norms for $H^1_{\Delta_N}(\mathbb{R}^n)$ using radial and non-tangential maximal functions, Riesz transforms, and atomic decompositions.
• To characterize $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ via the boundedness of commutators with Riesz transforms and the action of these transforms on $L^\infty$ functions.
• To prove that $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ strictly contains the classical $\text{BMO}(\mathbb{R}^n)$, and $H^1_{\Delta_N}(\mathbb{R}^n)$ is a proper subspace of $H^1(\mathbb{R}^n)$.
• To extend the duality between $H^1$ and $\text{BMO}$ to the Neumann Laplacian setting using weak factorizations and commutator estimates.

Proposed method

• Define $H^1_{\Delta_N}(\mathbb{R}^n)$ via the Littlewood–Paley area function associated with the Neumann Laplacian $\Delta_N$.
• Compute the Riesz transforms $R_{N,l} = \partial_l \Delta_N^{-1/2}$ as an additive perturbation of classical Riesz transforms.
• Establish weak factorization of $H^1_{\Delta_N}(\mathbb{R}^n)$ using products of $L^2$ functions via the Neumann Laplacian's Riesz transforms.
• Use the weak factorization to relate the norm of $f \in H^1_{\Delta_N}(\mathbb{R}^n)$ to the operator norm of commutators $[b, R_{N,l}]$ on $L^2(\mathbb{R}^n)$.
• Decompose functions on $\mathbb{R}^n$ into even and odd extensions to relate $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ to classical $\text{BMO}(\mathbb{R}^n)$ via $b_{+,e}$ and $b_{-,e}$.
• Apply known commutator estimates from Coifman, Rochberg, and Weiss to control the norm of $[b, \Delta_N^{-\alpha/2}]$ in terms of $\|b\|_{\text{BMO}_{\Delta_N}}$.

Experimental results

Research questions

• RQ1 Can the Hardy space $H^1_{\Delta_N}(\mathbb{R}^n)$ be characterized equivalently via maximal functions, Riesz transforms, atoms, and weak factorizations?
• RQ2 How does the space $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ relate to the classical $\text{BMO}(\mathbb{R}^n)$, and can it be characterized via the action of Riesz transforms on $L^\infty$ functions?
• RQ3 Is the commutator $[b, R_{N,l}]$ bounded on $L^2(\mathbb{R}^n)$ if and only if $b \in \text{BMO}_{\Delta_N}(\mathbb{R}^n)$?
• RQ4 Can the fractional integral $[b, \Delta_N^{-\alpha/2}]$ be bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ with operator norm controlled by $\|b\|_{\text{BMO}_{\Delta_N}}$?
• RQ5 What is the precise relationship between $H^1_{\Delta_N}(\mathbb{R}^n)$ and the classical $H^1(\mathbb{R}^n)$, and how does the Neumann boundary condition affect the structure of atoms and decompositions?

Key findings

• The Hardy space $H^1_{\Delta_N}(\mathbb{R}^n)$ admits equivalent norms defined via the radial maximal function, non-tangential maximal function, Riesz transforms, and atomic decomposition associated with the Neumann Laplacian.
• The space $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ is characterized by the boundedness of the commutator $[b, R_{N,l}]$ on $L^2(\mathbb{R}^n)$, with $\|b\|_{\text{BMO}_{\Delta_N}} \approx \|[b, R_{N,l}]: L^2 \to L^2\|$.
• $\text{BMO}_{\Delta_N}(\mathbb{R}^n)$ strictly contains $\text{BMO}(\mathbb{R}^n)$, as shown by the decomposition $b = b_{+,e} + b_{-,e}$ with $\|b\|_{\text{BMO}_{\Delta_N}} \approx \|b_{+,e}\|_{\text{BMO}} + \|b_{-,e}\|_{\text{BMO}}$.
• The commutator $[b, \Delta_N^{-\alpha/2}]$ is bounded from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ with operator norm controlled by $\|b\|_{\text{BMO}_{\Delta_N}}$, extending classical results to the Neumann setting.
• Weak factorization of $H^1_{\Delta_N}(\mathbb{R}^n)$ via $\Pi_l(g,h) = \sum_{j,k} \lambda_j^k g_j^k h_j^k$ allows the duality between $H^1_{\Delta_N}$ and $\text{BMO}_{\Delta_N}$ to be expressed through commutator norms.
• The Riesz transforms $R_{N,l}$ associated with $\Delta_N$ are an additive perturbation of the classical Riesz transforms, enabling the transfer of classical harmonic analysis techniques to the Neumann setting.