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Let M be a real-analytic submanifold of ${ℂ}^n$ whose “microlocal” Levi form has constant rank $s_M^{+}+s_M^{-}$ in a neighborhood of a prescribed conormal. Then local non-solvability of the tangential ∂̅-system is proved for forms of degrees $s_M^{-}$, $s_M^{+}$ (and 0).  This phenomenon is known in the literature as “absence of the Poincaré Lemma” and was already proved in case the Levi form is non-degenerate (i.e. $s_M^{-}+s_M^{+}=n-codimM$). We owe its proof to [2] and [1] in the case of a hypersurface and of a higher-codimensional submanifold respectively....

For a wedge $W$ of ${\mathbb{C}}^N$, we introduce two conditions of weak $q$-pseudoconvexity, and prove that they entail solvability of the $\overline{\delta }$-system for forms of degree $\ge q+1$ with coefficients in $C^{\mathrm{\infty }}\left(W\right)$ and $C^{\mathrm{\infty }}\left(\overline{W}\right)$ respectively. Existence and regularity for $\overline{\delta }$ in $W$ is treated by Hörmander [5, 6] (and also by Zampieri [9, 11] in case of piecewise smooth boundaries). Regularity in $W$ is treated by Henkin [4] (strong $q$-pseudoconvexity by the method of the integral representation), Dufresnoy [3] (full pseudoconvexity), Michel [8] (constant...

Si discute l'esistenza di soluzioni ${\mathrm{\Gamma }}^{(d)}$ su insiemi aperti $\mathrm{\Omega }\subset {ℝ}^n$ per equazioni differenziali iperbolico-ipoellittiche. Si dà una caratterizzazione geometrica quasi completa per aperti $\mathrm{\Omega }\subset {ℝ}^2$.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

Si discute l'esistenza di soluzioni ${\mathrm{\Gamma }}^{(d)}$ su insiemi aperti $\mathrm{\Omega }\subset {ℝ}^n$ per equazioni differenziali iperbolico-ipoellittiche. Si dà una caratterizzazione geometrica quasi completa per aperti $\mathrm{\Omega }\subset {ℝ}^2$.

Sia $K\subset \subset {ℝ}^n$ un compatto, $f\ge 0$ una funzione analitica all'intorno di $K$, ed $m$ la massima molteplicità in $K$ degli zeri di $f$; si prova che la potenza $f^{\lambda }$ ($\lambda \in ℂ$, $Re\lambda >\mathrm{—}\frac{1}{m}$) è integrabile in $K$. L'estensione meromorfa dell'applicazione $\lambda \to f^{\lambda }$ da $Re\lambda >0$ a tutto $ℂ$ (con valori in ${𝒟}^{\prime }(K)$ anziché in $L^1(K)$) era già stata provata in [1] e [2].

We prove that for a real analytic generic submanifold $M$ of ${\mathbb{C}}^n$ whose Levi-form has constant rank, the tangential ${\overline{\partial }}_M$-system is non-solvable in degrees equal to the numbers of positive and $M$ negative Levi-eigenvalues. This was already proved in [1] in case the Levi-form is non-degenerate (with $M$ non-necessarily real analytic). We refer to our forthcoming paper [7] for more extensive proofs.

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...