We prove the analyticity of $q$-concave sets of locally finite Hausdorff $(2n-2q)$-measure in a $n$-dimensional complex space. We apply it to give a removability criterion for meromorphic maps with values in $q$-complete spaces.

It is proved that if $M$ is a weakly 1-complete Kähler manifold with only one end, then $H_c^1(M,𝒪)=0$ or there exists a proper holomorphic mapping of $M$ onto a Riemann surface.

Si dimostra un risultato di prolungamento per applicazioni meromorfe a valori in uno spazio $q$-completo che generalizza direttamente il risultato classico di Hartogs e migliora risultati di K. Stein.

Let $X$ be a Stein manifold of complex dimension $n\ge 2$ and $\Omega ⋐X$ be a relatively compact domain with $C^2$ smooth boundary in $X$. Assume that $\Omega$ is a weakly $q$-pseudoconvex domain in $X$. The purpose of this paper is to establish sufficient conditions for the closed range of $\overline{\partial }$ on $\Omega$. Moreover, we study the $\overline{\partial }$-problem on $\Omega$. Specifically, we use the modified weight function method to study the weighted $\overline{\partial }$-problem with exact support in $\Omega$. Our method relies on the $L^2$-estimates by Hörmander (1965) and by Kohn (1973).

In this paper we extend to complex spaces and coherent analytic sheaves some results of Andreotti and Norguet concerning the extension of cohomology classes.

On a bounded $q$-pseudoconvex domain $\Omega$ in ${ℂ}^n$ with a Lipschitz boundary, we prove that the $\overline{\partial }$-Neumann operator $N$ satisfies a subelliptic $(1/2)$-estimate on $\Omega$ and $N$ can be extended as a bounded operator from Sobolev $(-1/2)$-spaces to Sobolev $(1/2)$-spaces.