Boundedness of Linear Operators via Atoms on Hardy Spaces with Non-doubling Measures

Abstract

Let $\mu$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(\mu)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(\mu)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, \gamma)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(\mu)$ to $L^{1, \infty}(\mu)$, if $T$ maps all $(p, \gamma)$-atomic blocks with $p\in(1, \infty)$ and $\gamma\in{\mathbb N}$ into uniformly bounded elements of $L^1(\mu)$, then $T$ extends to a bounded sublinear operator from $H^1(\mu)$ to $L^1(\mu)$. For the localized atomic Hardy space $h^1(\mu)$, corresponding results are also presented. Finally, these results are applied to Calder\'on-Zygmund operators, Riesz potentials and multilinear commutators generated by Calder\'on-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.