Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

TL;DR

This paper develops a frame decomposition for Hardy spaces associated with certain operators and characterizes these spaces using radial maximal functions, extending harmonic analysis tools to more general settings.

Contribution

It introduces a new frame decomposition for Hardy spaces linked to operators with Gaussian bounds and characterizes these spaces via radial maximal functions.

Findings

Established frame decomposition for Hardy spaces $H_{L}^{1}$

Proved norm equivalence via frame coefficients

Characterized $H_{L}^{1}$ using radial maximal semigroup

Abstract

Let $L$ be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and H\"older's continuity. Also assume that $L$ has a bounded holomorphic functional calculus on $L^2(\mathbb{R}^n)$. In this paper, we construct a frame decomposition for the functions belonging to the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ associated to $L$, and for functions in the Lebesgue spaces $L^p$, $1<p<\infty$. We then show that the corresponding $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of a function $f$ in terms of the frame coefficients is equivalent to the $H_{L}^{1}(\mathbb{R}^n)$-norm (resp. $L^p(\mathbb{R}^n)$-norm) of $f$. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space $H_{L}^{1}(\mathbb{R}^n)$ under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of $L$.