Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hardy space of exact forms on $\mathbb{R}^N$

Authors: Zengjian Lou and Alan McIntosh
Journal: Trans. Amer. Math. Soc. 357 (2005), 1469-1496
MSC (2000): Primary 42B30; Secondary 35J45, 58A10
Published electronically: September 2, 2004
MathSciNet review: 2115373
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Abstract: We show that the Hardy space of divergence-free vector fields on $\mathbb{R}^{3}$ has a divergence-free atomic decomposition, and thus we characterize its dual as a variant of $BMO$. Using the duality result we prove a ``div-curl" type theorem: for $b$ in $L^{2}_{loc}(\mathbb{R}^{3}, \mathbb{R}^{3})$, $\sup \int b\cdot (\nabla u\times \nabla v) dx$ is equivalent to a $BMO$-type norm of $b$, where the supremum is taken over all $u, v\in W^{1,2}(\mathbb{R}^{3})$ with $\Vert\nabla u\Vert _{L^{2}}, \Vert\nabla v\Vert _{L^{2}}\le 1.$ This theorem is used to obtain some coercivity results for quadratic forms which arise in the linearization of polyconvex variational integrals studied in nonlinear elasticity. In addition, we introduce Hardy spaces of exact forms on $\mathbb{R}^N$, study their atomic decompositions and dual spaces, and establish ``div-curl" type theorems on $\mathbb{R}^N$.

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Additional Information

Zengjian Lou
Affiliation: Institute of Mathematics, Shantou University, Shantou Guangdong 515063, People’s Republic of China

Alan McIntosh
Affiliation: Center for Mathematics and its Applications, Mathematical Sciences Institute, the Australian National University, Canberra, Australian Capital Territory 0200, Australia

Keywords: Divergence-free Hardy space, Hardy space of exact forms, atomic decomposition, $BMO$, div-curl, coercivity
Received by editor(s): May 16, 2003
Received by editor(s) in revised form: October 19, 2003
Published electronically: September 2, 2004
Additional Notes: The authors are supported by the Australian Government through the Australian Research Council. This paper was written when both authors were at the Center for Mathematics and its Applications of the Mathematical Sciences Institute at the Australian National University.
Article copyright: © Copyright 2004 American Mathematical Society