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Abstract

Let $({\mathcal X}, d, \mu)$ be a metric measure space of homogeneous type in the sense of Coifman and Weiss, with an “upper dimension” constant $n$. Let $p\in(n/(n+1),1)$. For any $f$ in the Hardy space $H_{\rm at}^p({\mathcal X})$ and $g$ in the Lipschitz space $\operatorname{Lip}_{1/p-1}({\mathcal X})$, the authors prove that the product of $f$ and $g$, viewed as a distribution, can be decomposed into $\mathcal L(f,g)+\mathcal H(f,g)$, where $\mathcal L$ is a bilinear operator bounded from $H_{\rm at}^p({\mathcal X})\times \operatorname{Lip}_{1/p-1}({\mathcal X})$ to $L^1({\mathcal X})$ and $\mathcal H$ a bilinear operator bounded from $H_{\rm at}^p({\mathcal X})\times \operatorname{Lip}_{1/p-1}({\mathcal X})$ to a Hardy space of Musielak–Orlicz type, $H^{\Xi_p}({\mathcal X})$. Moreover, when ${\mathcal X}$ is the Euclidean space, this bilinear decomposition is sharp in the sense that the space $H^{\Xi_p}({\mathcal X})$ cannot be replaced by a smaller space. As applications, if $b\in\operatorname{Lip}_{1/p-1}({\mathcal X})$ is nonconstant and $T$ a Calderón–Zygmund operator, the authors find the largest subspace of $H_{\rm at}^p({\mathcal X})$, denoted by $H_b^p({\mathcal X})$, such that the commutator $[b, T]$ is bounded from $H_b^p({\mathcal X})$ to $L^1({\mathcal X})$, or to $H_{\rm at}^1({\mathcal X})$ when $T$ further satisfies some cancelation conditions. The novelty of these results is that the underlying measure $\mu$ is only assumed to be doubling.