Regularization of $L^ 2$ norms of Lagrangian distributions

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.