Bilinear decompositions of products of local Hardy and Lipschitz or BMO spaces through wavelets

Let p is an element of (n/n+1, 1] and f is an element of h(p)(R-n) be the local Hardy space in the sense of D. Gold-berg. In this paper, the authors establish two bilinear decompositions of the product spaces of h(p)(R-n) and their dual spaces. More precisely, the authors prove that h(1)(R-n) x bmo(R-n) = L-1(R-n) + h(*)(Phi) (R-n) and, for any p is an element of (n/n+ 1, 1), h(p)(R-n) x Lambda(alpha)(R-n) = L-1(R-n) + h(p)(R-n), where bmo((R)n) denotes the local BMO space, Lambda(alpha)(R-n), for any p is an element of (n/n+ 1, 1) and alpha := n(1/p - 1), the inhomogeneous Lipschitz space and hF* (Rn) a variant of the local Orlicz-Hardy space related to the Orlicz function Phi(t) := t/log(e+ t) for any t is an element of [ 0, infinity) which was introduced by Bonami and Feuto. As an application, the authors establish a div-curl lemma at the endpoint case.