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Let $A$ be a closed set of $M\simeq {\mathbb{R}}^n$, whose conormai cones $x+y_x^{*}\left(A\right)$, $x\in A$, have locally empty intersection. We first show in §1 that $\text{dist}\left(x,A\right)$, $x\in M\setminus A$ 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 ${\left.{\mathcal{C}}_{A|X}\right|}_{{\dot{\pi }}^{-1}\left(x_0\right)}$, $x_0\in \partial A$, satisfy the principle of the analytic continuation in the complex integral manifolds of ${\left\{H\left({\varphi }_i^C\right)\right\}}_{i=1,\mathrm{\dots },m}$, $\left\{{\varphi }_i\right\}$ being a base for the linear hull of ${\gamma }_{x_0}^{*}\left(A\right)$ in $T_{x_0}^{*}M$; in particular we get ${\left.{\mathrm{\Gamma }}_{A\times {}_{M}T{}^{*}{}_{M}X}\left({\mathcal{C}}_{A|X}\right)\right|}_{\partial A\times {}_M\dot{T}{}^{*}{}_{M}X}=0$. When $A$is a half space with $C^{\omega }$-boundary,...

Let $X\cong C^n$, let $M$ be a $C^2$ hypersurface of $X$, $S$ be a $C^2$ submanifold of $M$. Denote by $L_M$ the Levi form of $M$ at $z_0\in S$. In a previous paper [3] two numbers $s^{\pm }\left(S,p\right)$, $p\in {\left({\dot{T}}_S^{*}X\right)}_{z_0}$ are defined; for $S=M$ they are the numbers of positive and negative eigenvalues for $L_M$. For $S\subset M$, $p\in S\times {}_M\dot{T}{}^{*}{}_{S}X)$, we show here that $s^{\pm }\left(S,p\right)$ are still the numbers of positive and negative eigenvalues for $L_M$ when restricted to $T_{z_0}^{C}S$. Applications to the concentration in degree for microfunctions at the boundary are given.

Let $X$ be a complex manifold, $S$ a generic submanifold of $X^{\mathbb{R}}$, the real underlying manifold to $X$. Let $\mathrm{\Omega }$ be an open subset of $S$ with $\partial \mathrm{\Omega }$ analytic, $Y$ a complexification of $S$. We first recall the notion of $\mathrm{\Omega }$-tuboid of $X$ and of $Y$ and then give a relation between; we then give the corresponding result in terms of microfunctions at the boundary. We relate the regularity at the boundary for ${\overline{\partial }}_b$ to the extendability of $CR$ functions on $\mathrm{\Omega }$ to $\mathrm{\Omega }$-tuboids of $X$. Next, if $X$ has complex dimension 2, we give results on extension...