Paraproducts and Products of functions in $BMO({\mathbb{R}}^n)$ and $H^1({\mathbb{R}}^n)$ through wavelets

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $\text{BMO}({\text{bR}}^n)$ and \H^1(\bR^n), may be written as the sum of two continuous bilinear operators, one from \H^1(\bR^n)\times \BMO(\bR^n) into $L^1({\text{bR}}^n)$, the other one from \H^1(\bR^n)\times \BMO(\bR^n) into a new kind of Hardy-Orlicz space denoted by \H^{\log}(\bR^n). More precisely, the space \H^{\log}(\bR^n) is the set of distributions $f$ whose grand maximal function $\mathcal{M}f$ satisfies $${\int }_{{\mathbb{R}}^n}\frac{\mathrm{\mid }\mathcal{M}f(x)\mathrm{\mid }}{\log (e+\mathrm{\mid }x\mathrm{\mid })+\log (e+\mathrm{\mid }\mathcal{M}f(x)\mathrm{\mid })}dx<\mathrm{\infty }\mathrm{.}$$ The two bilinear operators can be defined in terms of paraproducts. As a consequence, we find an endpoint estimate involving the space \H^{\log}(\bR^n) for the $÷$-$\text{curl}$ lemma.