Paraproducts and products of functions in [formula omitted] and [formula omitted] through wavelets

Abstract

In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in BMO(Rn) and H1(Rn), may be written as the sum of two continuous bilinear operators, one from H1(Rn)×BMO(Rn) into L1(Rn), the other one from H1(Rn)×BMO(Rn) into a new kind of Hardy–Orlicz space denoted by Hlog(Rn). More precisely, the space Hlog(Rn) is the set of distributions f whose grand maximal function Mf satisfies∫Rn|Mf(x)|log(e+|x|)+log(e+|Mf(x)|)dx<∞. The two bilinear operators can be defined in terms of paraproduct. As a consequence, we find an endpoint estimate involving the space Hlog(Rn) for the div-curl lemma.