Bilinear decomposition and divergence-curl estimates on products related to local Hardy spaces and their dual spaces

Let p∈(0,1), α:=1/p−1, and, for any τ∈[0,∞), Φp(τ):=τ/(1+τ1−p). Hp(Rn), hp(Rn), and Λnα(Rn) be, respectively, the Hardy space, local inhomogeneous Lipschitz space on Rn. In this article, applying renormalization of wavelets, authors establish a bilinear decomposition multiplications elements in hp(Rn) [or Hp(Rn)] Λnα(Rn), prove that these decompositions are sharp some sense. As applications, also obtain estimates product with p∈(0,1] its dual zero ⌊nα⌋-inhomogeneous curl divergence, where ⌊nα⌋ denotes largest integer not greater than nα. Moreover, find new structures hΦp(Rn) HΦp(Rn) by showing hΦp(Rn)=h1(Rn)+hp(Rn) HΦp(Rn)=H1(Rn)+Hp(Rn) equivalent quasi-norms, spaces both coincide. These results give complete picture multiplication between space.