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 $\mathcal{M}f$ satisfies $${\int }_{{\mathbb{R}}^n}\frac{|\mathcal{M}f(x)|}{\log (e+|x|)+\log (e+|\mathcal{M}f(x)|)}dx<\infty .$$ 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 $÷$-$\curl$ lemma.