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# Regularization of L2 Norms of Lagrangian Distributions

Steven Izen
Transactions of the American Mathematical Society
Vol. 288, No. 1 (Mar., 1985), pp. 363-380
DOI: 10.2307/2000444
Stable URL: http://www.jstor.org/stable/2000444
Page Count: 18
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## Abstract

Let X be a compact smooth manifold, dim X = n. Let Λ be a fixed Lagrangian submanifold of $T^\astX$. The space of Lagrangian distributions Ik(X, Λ) is contained in L2(X) if $k < -n/4$. When k = n/4, I-n/4(X, Λ) just misses L2(X). A new inner product $\langle u, \upsilon\rangle_R$ is defined on I-n/4(X, Λ)/I-n/4-1(X, Λ) in terms of symbols. This inner product contains "L2 information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in L2(X). The asymptotic behavior of the L2 inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case, $\langle u, \upsilon\rangle_R$ is found to regulate the growth of the ordinary L2 inner product.

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