A propagation theorem for a class of microfunctions

D'Agnolo, Andrea, and Zampieri, Giuseppe. "A propagation theorem for a class of microfunctions." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 1.1 (1990): 53-58. <http://eudml.org/doc/244098>.

@article{DAgnolo1990,
abstract = {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 \( \mathcal\{M\} \).},
author = {D'Agnolo, Andrea, Zampieri, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Partial differential equations on manifolds; Boundary value problems; Theory of functions; unique continuation theorem; microfunction; micro-analyticity; wave-front set},
language = {eng},
month = {2},
number = {1},
pages = {53-58},
publisher = {Accademia Nazionale dei Lincei},
title = {A propagation theorem for a class of microfunctions},
url = {http://eudml.org/doc/244098},
volume = {1},
year = {1990},
}

TY - JOUR
AU - D'Agnolo, Andrea
AU - Zampieri, Giuseppe
TI - A propagation theorem for a class of microfunctions
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1990/2//
PB - Accademia Nazionale dei Lincei
VL - 1
IS - 1
SP - 53
EP - 58
AB - 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 \( \mathcal{M} \).
LA - eng
KW - Partial differential equations on manifolds; Boundary value problems; Theory of functions; unique continuation theorem; microfunction; micro-analyticity; wave-front set
UR - http://eudml.org/doc/244098
ER -