A propagation theorem for a class of microfunctions

D'Agnolo, Andrea ; Zampieri, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 1 (1990) no. 1, pp. 53-58.

Abstract (VO)

Let \( A \) be a closed set of \( M \simeq \mathbb{R}^{n} \), whose conormai cones \( x + y^{*}_{x}(A) \), \( x \in A \), have locally empty intersection. We first show in §1 that \( \text{dist}(x,A) \), \( x \in M \setminus A \) is a \( C^{1} \) function. We then represent the n microfunctions of \( \mathcal{C}_{A|X} \), \( X \simeq \mathbb{C}^{n} \), using cohomology groups of \( \mathcal{O}_{X} \) of degree 1. By the results of § 1-3, we are able to prove in §4 that the sections of \( \mathcal{C}_{A|X}\large|_{\dot{\pi}^{-1}(x_{0})} \), \( x_{0} \in \partial A \), satisfy the principle of the analytic continuation in the complex integral manifolds of \( \{H(\phi_{i}^{C})\}_{i=1, \ldots, m} \), \( \{\phi_{i}\} \) being a base for the linear hull of \( \gamma^{*}_{x_{0}}(A) \) in \( T^{*}_{x_{0}}M \); in particular we get \( \Gamma_{A \times_{M} T^{*}_{M}X}(\mathcal{C}_{A|X})\large|_{\partial A \times_{M} \dot{T}^{*}_{M}X} = 0 \). When \( A \)is a half space with \( C^{\omega} \)-boundary, all of the above results werealready proved by Kataoka. Finally for a \( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \)\( \mathcal{E}_{X} \)-module \( \mathcal{M} \) we show that \( \mathcal{H}\mathit{om}_{\mathcal{E}_{X}}(\mathcal{M}, \mathcal{C}_{A|X})_{p} = 0 \), when at least one conormal \( \theta \in \dot{\gamma}^{*}_{x_{0}}(A) \) is non-characteristic for \( \mathcal{M} \).

Riassunto

Sia \( A \) un insieme chiuso di \( M \simeq \mathbb{R}^{n} \) i cui coni conormali \( x + y^{*}_{x}(A) \), \( x \in A \), hanno localmente intersezione vuota. Si prova nel §1 che \( \text{dist}(x,A) \), \( x \in M \setminus A \) è una funzione \( C^{1} \). Si rappresentano poi le microfunzioni di \( \mathcal{C}_{A|X} (X \simeq \mathbb{C}^{n} )\), mediante gruppi di coomologia di \( \mathcal{O}_{X} \) in grado 1. Se ne deduce nel §4 un principio di prolungamento analitico per sezioni di \( \mathcal{C}_{A|X}\large|_{\dot{\pi}^{-1}(x_{0})} \), \( x_{0} \in \partial A \) che generalizza alcuni risultati di Kataoka. Se ne dà infine applicazione ai problemi ai limiti.

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