Hardy Spaces $H_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates

TL;DR

This paper introduces Hardy spaces associated with operators satisfying $k$-Davies-Gaffney estimates, characterizes them via molecules and square functions, and proves boundedness of related Riesz transforms, extending known results for specific cases.

Contribution

It develops a new framework for Hardy spaces linked to operators with $k$-Davies-Gaffney estimates, including molecular and square function characterizations, and analyzes Riesz transform boundedness.

Findings

Defined Hardy spaces $H_L^p$ for operators with $k$-Davies-Gaffney estimates.

Proved molecular and square function characterizations of these Hardy spaces.

Established boundedness of Riesz transforms from $H_L^p$ to classical Hardy spaces.

Abstract

Let $L$ be a one to one operator of type $\omega$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the Hardy space $H_L^p(\mathbb{R}^n)$ with $p\in (0,\,1]$ associated to $L$ in terms of square functions defined via $\{e^{-t^{2k}L}\}_{t>0}$ and establish their molecular and generalized square function characterizations. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1$ with complex bounded measurable coefficients and the $2k$-order Schr\"odinger type operator $L_2\equiv (-\Delta)^k+V^k$, where $\Delta$ is the Laplacian and $0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$. Moreover, as applications, for $i\in\{1,\,2\}$, the authors prove that the associated Riesz transform $\nabla^k(L_i^{-1/2})$ is bounded from $H_{L_i}^p(\mathbb{R}^n)$ to $H^p(\mathbb{R}^n)$ for $p\in(n/(n+k),\,1]$ and establish the Riesz transform characterizations of $H_{L_1}^p(\mathbb{R}^n)$ for $ p\in(rn/(n+kr),\,1]$ if $\{e^{-tL_1}\}_{t>0}$ satisfies the $L^r-L^2$ $k$-off-diagonal estimates with $r\in (1,2]$. These results when $k\equiv1$ and $L\equiv L_1$ are known.