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Let $({\mathcal X},d,\mu)$ be a metric measure space of homogeneous type inthe sense of R. R. Coifman and G. Weiss and $H^1_{\rm at}({\mathcal X})$ be theatomic Hardy space. Via orthonormal bases of regular wavelets and splinefunctions recently constructed by P. Auscher and T. Hyt\"onen, the authorsprove that the product $f\times g$ of $f\in H^1_{\rm at}({\mathcal X})$ and$g\in\mathop\mathrm{BMO}({\mathcal X})$, viewed as a distribution, can bewritten into a sum of two bounded bilinear operators, respectively, from$H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ into$L^1({\mathcal X})$ and from $H^1_{\rm at}({\mathcalX})\times\mathop\mathrm{BMO}({\mathcal X})$ into $H^{\log}({\mathcal X})$,which affirmatively confirms the conjecture suggested by A. Bonami and F.Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl.425 (2015), 807-817]).

Products of Functions in ${\mathop\mathrm{BMO}}({\mathcal X})$ and $H^1_{\rm at}({\mathcal X})$ via Wavelets over Spaces of Homogeneous Type