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On the Kähler form of complex \(L^{p}\) space and its Lagrangian subspaces

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Abstract

In this article, we obtain the Kähler forms for complex \(L^{p}\) spaces, \(1\le p<\infty \), and we find and describe explicitly the set of all Lagrangian subspaces of the complex \(L^{p}\) space. The results in this article show that the Lagrangians of complex \(L^{2}\) space are distinct from those of complex \(L^{p}\) spaces for \(1\le p<\infty \), \(p\ne 2\). As an application, the symplectic structure determined by the Kähler form can be used to determine the symplectic form of the complex Holmes–Thompson volumes restricted on complex lines in integral geometry of complex \(L^{p}\) space.

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Acknowledgments

The author would like to thank Prof. J. Fu for some helpful discussions in this subject. This work was partially supported by NSF.

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  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Yang Liu

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  1. Yang Liu
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Correspondence to Yang Liu.

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Liu, Y. On the Kähler form of complex \(L^{p}\) space and its Lagrangian subspaces. J. Pseudo-Differ. Oper. Appl. 6, 265–277 (2015). https://doi.org/10.1007/s11868-015-0114-z

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  • DOI: https://doi.org/10.1007/s11868-015-0114-z

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