Bonami–Bernicot bilinear decomposition conjecture for products of BMO and H^1 functions

Let ${\mathcal X}$ be the underlying space of homogeneous type, and let $BMO({\mathcal X})$, $H^1({\mathcal X})$, $L^1({\mathcal X})$, and $H^{\log}({\mathcal X})$ denote the spaces used above. For $b\in BMO({\mathcal X})$ and $f\in H^1({\mathcal X})$, write $b\times f$ for their distributional product. Bonami–Bernicot conjecture. There exist two bounded bilinear operators

$$\mathscr L: BMO({\mathcal X})\times H^1({\mathcal X}) \to L^1({\mathcal X})$$

and

$$\mathscr H: BMO({\mathcal X})\times H^1({\mathcal X})\to H^{\log}({\mathcal X})$$

such that

$$b\times f = \mathscr L(b,f) + \mathscr H(b,f).$$

The preceding theorems establish analogous decompositions with operators depending linearly on a fixed Hardy-space function, while this conjecture asks for a jointly bounded bilinear decomposition. It is presented as a conjecture suggested by A. Bonami and F. Bernicot; no resolution is given in the supplied text.