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In this paper, we prove that the product (in the distribution sense) of twofunctions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may bewritten as the sum of two continuous bilinear operators, one from$\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from$\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denotedby $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the setof distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies$$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcalMf(x)|)}dx <\infty.$$ The two bilinear operators can be defined in terms ofparaproducts. As a consequence, we find an endpoint estimate involving thespace $\H^{\log}(\bR^n)$ for the $\div$-$\curl$ lemma.

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