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

Abstract

Let $$({\mathcal X},d,\mu )$$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $$H^1_\mathrm{at}({\mathcal X})$$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytonen, the authors prove that the product $$f\times g$$ of $$f\in H^1_\mathrm{at}({\mathcal X})$$ and $$g\in \mathrm {BMO}({\mathcal X})$$ , viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $$H^1_\mathrm{at}({\mathcal X})\times \mathrm {BMO}({\mathcal X})$$ into $$L^1({\mathcal X})$$ and from $$H^1_\mathrm{at}({\mathcal X}) \times \mathrm {BMO}({\mathcal X})$$ into $$H^{\log }({\mathcal X})$$ , which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by Ky in J Math Anal Appl 425:807–817, 2015).