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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Regularization of $L^ 2$ norms of Lagrangian distributions
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by Steven Izen PDF
Trans. Amer. Math. Soc. 288 (1985), 363-380 Request permission

Abstract:

Let $X$ be a compact smooth manifold, $\dim X = n$. Let $\Lambda$ be a fixed Lagrangian submanifold of ${T^\ast }X$. The space of Lagrangian distributions ${I^k}(X,\Lambda )$ is contained in ${L^2}(X)$ if $k < - n/4$. When $k = n/4$, ${I^{ - n/4}}(X,\Lambda )$ just misses ${L^2}(X)$. A new inner product ${\langle u,v\rangle _R}$ is defined on ${I^{ - n/4}}(X,\Lambda )/{I^{ - n/4 - 1}}(X,\Lambda )$ in terms of symbols. This inner product contains "${L^2}$ information" in the following sense: Slight regularizations of the Lagrangian distributions are taken, putting them in ${L^2}(X)$. The asymptotic behavior of the ${L^2}$ inner product is examined as the regularizations approach the identity. Three different regularization schemes are presented and, in each case, ${\langle u,v\rangle _R}$ is found to regulate the growth of the ordinary ${L^2}$ inner product.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 288 (1985), 363-380
  • MSC: Primary 58G15
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0773065-1
  • MathSciNet review: 773065