[Paper Review] A new proof of the atomic decomposition of Hardy spaces

Shai Dekel, G. Kerkyacharian+2 more|arXiv (Cornell University)|Sep 1, 2014

Advanced Harmonic Analysis Research6 references3 citations

TL;DR

This paper presents a new proof of the atomic decomposition of Hardy spaces $H^p$ for $0 < p \leq 1$ on $\mathbb{R}^n$, avoiding the classical Calderón-Zygmund decomposition and approximation of identity. Instead, it uses a novel decomposition based on smooth, compactly supported functions with vanishing moments and dyadic frequency localization, yielding a robust method applicable to general metric measure spaces with doubling and Gaussian heat kernel bounds.

ABSTRACT

A new proof is given of the atomic decomposition of Hardy spaces Hp, in the classical setting of Rn. The new method can be used to establish atomic decomposition of maximal Hardy spaces in general setting and non classical settings.

Motivation & Objective

• To provide a new, alternative proof of the classical atomic decomposition of $H^p$ spaces for $0 < p \leq 1$ on $\mathbb{R}^n$, avoiding reliance on Calderón-Zygmund decomposition.
• To develop a method that generalizes beyond the classical setting, particularly to nonclassical and abstract metric measure spaces.
• To establish the equivalence of maximal and atomic Hardy spaces in general settings, such as those with doubling measures and operators with Gaussian heat kernels.
• To demonstrate that the new method yields a constructive atomic decomposition with uniform control on the atomic norm in terms of the maximal function norm.
• To provide a framework that simplifies the atomic decomposition process by using smooth, compactly supported functions with vanishing moments and frequency localization.

Proposed method

• Construct a smooth, compactly supported function $\varphi$ with $\widehat{\varphi}(0) = 1$ and $\partial^\alpha \widehat{\varphi}(0) = 0$ for $0 < |\alpha| \leq m$, using a difference operator on a symmetric bump function.
• Define a dyadic frequency decomposition using $\psi_k(x) = 2^{kn}\psi(2^k x)$, where $\psi$ is a smooth function with compact Fourier support and vanishing moments up to order $K$.
• Introduce the maximal function $M^*_{\varphi,a}f(x)$ and the grand maximal operator $\mathcal{M}_N f(x)$, showing their equivalence to the Poisson maximal function in $L^p$-norm.
• Define the sets $\Omega_r = \{x : \sup_{t>0} |P_t * f(x)| > 2^r\}$ and decompose $f$ as $f = \sum_r F_r$, where $F_r$ is localized to $\Omega_r$.
• For each dyadic level $r$, decompose $F_r$ into pieces $F_B$ supported on dyadic balls $B \in \mathcal{B}_r$, using a Whitney-type covering of $\Omega_r$.
• Construct atoms $a_B$ via $a_B(x) = c_\sharp^{-1} |B^\star|^{-1/p} 2^{-r} F_B(x)$, with $B^\star = 7B$, and coefficients $\lambda_B = c_\sharp |B^\star|^{1/p} 2^r$, ensuring $L^\infty$ and moment conditions.

Experimental results

Research questions

• RQ1 Can the atomic decomposition of $H^p$ spaces be established without relying on the Calderón-Zygmund decomposition or approximation of identity?
• RQ2 What structural properties of the decomposition allow generalization to nonclassical settings such as metric measure spaces with doubling and Gaussian heat kernels?
• RQ3 How can the atomic norm $\|f\|_{H^p_A}$ be controlled by the maximal function norm $\|f\|_{H^p}$ using a frequency-localized, moment-based construction?
• RQ4 Is it possible to construct atoms with uniform $L^\infty$ and moment conditions using smooth, compactly supported functions with vanishing moments?
• RQ5 What is the role of the grand maximal operator $\mathcal{M}_N f$ in characterizing $H^p$ and enabling the atomic decomposition in a general setting?

Key findings

• The new proof establishes the atomic decomposition of $H^p$ for $0 < p \leq 1$ without using the Calderón-Zygmund decomposition or approximation of identity.
• The method yields the inequality $\|f\|_{H^p_A} \leq c \|f\|_{H^p}$, where $c > 0$ depends only on $p$ and $n$, proving the continuous embedding $H^p \subset H^p_A$.
• The construction of atoms $a_B$ ensures $\|a_B\|_{L^\infty} \leq |B^\star|^{-1/p}$ and $\int x^\alpha a_B(x) dx = 0$ for $|\alpha| \leq n(p^{-1} - 1)$, satisfying the standard atomic conditions.
• The decomposition $f = \sum_{r} \sum_{B \in \mathcal{B}_r} \lambda_B a_B$ converges in $\mathcal{S}'$, and the sum $\sum_{r} \sum_{B \in \mathcal{B}_r} |\lambda_B|^p \leq c \|f\|_{H^p}^p$ holds, confirming the atomic norm control.
• The method is robust enough to be extended to general metric measure spaces with doubling property and operators with Gaussian heat kernels and Markov property, as shown in subsequent work [2].
• The proof avoids the use of $L^2$-based techniques and instead relies on frequency localization and moment conditions, making it adaptable to non-smooth or non-Riemannian settings.