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

Abstract

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