Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces of homogeneous type

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 Lips