Hardy space of exact forms on $\mathbb {R}^N$
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- by Zengjian Lou and Alan M$^{\mathrm {c}}$Intosh PDF
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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 $\|\nabla u\|_{L^{2}}, \ \|\nabla v\|_{L^{2}}\le 1.$ 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 $\mathbb {R}^N$.References
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Additional Information
- Zengjian Lou
- Affiliation: Institute of Mathematics, Shantou University, Shantou Guangdong 515063, People’s Republic of China
- Email: zjlou@stu.edu.cn
- Alan M$^{\mathrm {c}}$Intosh
- Affiliation: Center for Mathematics and its Applications, Mathematical Sciences Institute, the Australian National University, Canberra, Australian Capital Territory 0200, Australia
- Email: alan@maths.anu.edu.au
- 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.
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 1469-1496
- MSC (2000): Primary 42B30; Secondary 35J45, 58A10
- DOI: https://doi.org/10.1090/S0002-9947-04-03535-4
- MathSciNet review: 2115373