## Abstract

Let ${L}_{1},{L}_{2}$ be a pair of one-to-one commuting sectorial
operators such that each *L _{i}* for $i\in \{1,2\}$ satisfies
the

*m*order

_{i}*L*

^{2}off-diagonal estimates and ${m}_{1}\ge {m}_{2}>0$. Let ${H}_{{L}_{i}}^{p}\left({\mathbb{R}}^{n}\right)$, $i\in \{1,2\}$, and ${H}_{{L}_{1}+{\tilde{L}}_{2}}^{p}\left({\mathbb{R}}^{n}\right)$ be the Hardy spaces associated, respectively, to the operators

*L*and ${L}_{1}+{\tilde{L}}_{2}$, where ${\tilde{L}}_{2}:={L}_{2}^{{m}_{1}/{m}_{2}}$ is a fractional power of

_{i}*L*

_{2}. In this paper, the authors give out some real-variable properties of these Hardy spaces. More precisely, the authors first establish the bounded joint

*H*

_{∞}functional calculus in these Hardy spaces and prove that the abstract Riesz transform ${D}^{{m}_{i}}{({L}_{1}+{L}_{2})}^{-1/2}$ is bounded from ${H}_{{L}_{i}}^{p}\left({\mathbb{R}}^{n}\right)$ to the classical Hardy space ${H}^{p}\left({\mathbb{R}}^{n}\right)$ for all $p\in (\frac{n}{n+{m}_{i}},1]$, where $i\in \{1,2\}$. Moreover, for all $p\in (0,1]$, the authors show that ${H}_{{L}_{1}+{\tilde{L}}_{2}}^{p}\left({\mathbb{R}}^{n}\right)={H}_{{L}_{1}}^{p}\left({\mathbb{R}}^{n}\right)+{H}_{{L}_{2}}^{p}\left({\mathbb{R}}^{n}\right)$ and give a sufficient condition to guarantee ${H}_{{L}_{1}}^{p}\left({\mathbb{R}}^{n}\right)\subset {H}_{{L}_{2}}^{p}\left({\mathbb{R}}^{n}\right)$.

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