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

Let p is an element of (0,1), alpha:= 1/p - 1, and, for any tau is an element of [0, infinity), Phi(P)(tau) := tau/(1 + tau(1-p)). Let H-P(R-n), h(P)(R-n), and Lambda(n alpha) (R-n) be, respectively, the Hardy space, the local Hardy space, and the inhomogeneous Lipschitz space on R-n. In this article, applying the inhomogeneous renormalization of wavelets, the authors establish a bilinear decomposition for multiplications of elements in h(P)(R-n) [or H-P(R-n)] and Lambda(n alpha) (R-n), 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 h(P)(R-n) with p is an element of (0, 1] and its dual space, respectively, with zero left perpendicularn alpha right perpendicular-inhomogeneous curl and zero divergence, where left perpendicularn alpha right perpendicular denotes the largest integer not greater than ma. Moreover, the authors find new structures of h(Phi p)(R-n) and H-Phi p (R-n) by showing that h(Phi p)(R-n) = h(1)(R-n) + h(p)(R-n) and H-Phi p = h(1)(R-n) + H-p(R-n) with equivalent quasi-norms, and also prove that the dual spaces of both h(Phi p)(R-n) and h(p) (R-n) coincide. These results give a complete picture on the multiplication between the local Hardy space and its dual space. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

In this article, the authors introduce a bilinear decomposition for multiplications of elements in the local Hardy space and its dual space using the inhomogeneous renormalization of wavelets. They prove that these bilinear decompositions are sharp in some sense. Additionally, the authors obtain estimates for elements in the local Hardy space and show new structures of different spaces.