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

TL;DR

This paper demonstrates that the product of functions from atomic Hardy space and BMO space on a metric measure space can be decomposed into two bounded bilinear operators, confirming a conjecture in harmonic analysis.

Contribution

The authors prove a new decomposition of products of Hardy and BMO functions on spaces of homogeneous type using wavelet bases, confirming a conjecture by Bonami and Bernicot.

Findings

Product decomposes into two bounded bilinear operators

Operators map into L^1 and H^{log} spaces

Confirms a conjecture by Bonami and Bernicot

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 $H^1_{\rm at}({\mathcal X})$ be the atomic Hardy space. Via orthonormal bases of regular wavelets and spline functions recently constructed by P. Auscher and T. Hyt\"onen, the authors prove 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 be written 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 $H^1_{\rm at}({\mathcal X})\times\mathop\mathrm{BMO}({\mathcal X})$ 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]).