Paraproducts and Products of Functions in BMO(Rn) and H1(Rn) ThroughWavelets

As an important application of Musielak-Orlicz Hardy spaces, in this chapter, we prove that the product of two functions, which are respectively in BMO(R-n) and H-1((R-n), may be written as the sum of two continuous bilinear operators, one is bounded from H-1(R-n) x BMO(R-n) into L-1(R-n), the other one from H-1(R-n) x BMO(R-n) into a special Musielak-Orlicz Hardy space H-log(R-n), associated with the growth function theta(x,t) := t/log(e + vertical bar x vertical bar) + log(e + t)(.) The two bilinear operators can be defined in terms of paraproducts. As a further application of this, we find an endpoint estimate involving the space H-log(R-n) for the div-curl lemma.