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

Abstract

Let $μ$ 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 $H^1(μ)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(μ)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, γ)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(μ)$ to $L^{1, \infty}(μ)$, if $T$ maps all $(p, γ)$-atomic blocks with $p\in(1, \infty)$ and $γ\in{\mathbb N}$ into uniformly bounded elements of $L^1(μ)$, then $T$ extends to a bounded sublinear operator from $H^1(μ)$ to $L^1(μ)$. For the localized atomic Hardy space $h^1(μ)$, corresponding results are also presented. Finally, these results are applied to Calderón-Zygmund operators, Riesz potentials and multilinear commutators generated by Calderón-Zygmund operators or fractional integral operators with Lipschitz functions, to simplify the existing proofs in the corresponding papers.