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

Abstract

Let $(\mathcal{X},d,\hat{I}¼)$ be a metric measure space of homogeneous type in the sense of R. R. Coifman and G. Weiss and $H_{\mathrm{at}}^1(\mathcal{X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hytönen, the authors prove that the product $f\times g$ of $f\in H_{\mathrm{at}}^1(\mathcal{X})$ and $g\in \mathrm{BMO}(\mathcal{X})$, viewed as a distribution, can be written into a sum of two bounded bilinear operators, respectively, from $H_{\mathrm{at}}^1(\mathcal{X})\times \mathrm{BMO}(\mathcal{X})$ into $L^1(\mathcal{X})$ and from $H_{\mathrm{at}}^1(\mathcal{X})\times \mathrm{BMO}(\mathcal{X})$ into $H^{\log }(\mathcal{X})$, which affirmatively confirms the conjecture suggested by A. Bonami and F. Bernicot (This conjecture was presented by L. D. Ky in [J. Math. Anal. Appl. 425 (2015), 807-817]).

66 pages, Submitted