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Let $X$ be a complex analytic manifold, $M \subset X$ a $C^2$ submanifold, $\Omega\subset M$ an open set with $C^2$ boundary $S=\partial\Omega$. Denote by $\mu_M({\cal O}_X)$ (resp. $\mu_\Omega({\cal O}_X)$) the microlocalization along $M$ (resp. $\Omega$) of the sheaf ${\cal O}_X$ of holomorphic functions.
In the literature (cf. [A-G], [K-S 1,2]) one encounters two classical results concerning the vanishing of the cohomology groups$H^j\mu_M({\cal O}_X)_p$ for $p\in \dot{T}^*_MX$. The most general gives the vanishing outside a range of indices $j$ whose length is equal to $s^0(M,p)$ (with $s^{+,-,0}(M,p)$ being the number of respectively positive, negative and null eigenvalues for the ‘microlocal’ Levi form $L_M(p)$). The sharpest result gives the concentration in a single degree, provided that the difference $s^-(M,p^{\prime})-\gamma(M,p^{\prime})$ is locally constant for $p^{\prime}\in T^*_MX$ near $p$ (with $\gamma(M,p)=\dim^{\rm C}(T^*_MX\cap iT^*_MX)_z$ for $z$ the base point of $p$).
The first result was restated for the complex $\mu_\Omega({\cal O}_X)$ in [D'A-Z 2], in the case ${\rm codim}_MS=1$. We extend it here to any codimension and moreover we also restate for $\mu_\Omega({\cal O}_X)$ the second vanishing theorem.
We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.
