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

Abstract: In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\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 $L^1(\bR^n)$, 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 $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathcal Mf(x)|)}dx <\infty.$$ 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 $\div$-$\curl$ lemma.