$C^k$ estimates for convex domains of finite type in ${ℂ}^n$ are known from [7] for $k=0$ and from [2] for $k\>0$. We want to show the same result for concave domains of finite type. As in the case of strictly pseudoconvex domain, we fit the method used in the convex case to the concave one by switching $z$ and $\zeta$ in the integral kernel of the operator used in the convex case. However the kernel will not have the same behavior on the boundary as in the Diederich-Fischer-Fornæss-Alexandre work. To overcome this problem...

Following Sibony, we say that a bounded domain $\Omega$ in $C^n$ is $B$-regular if every continuous real valued function on the boundary of $\Omega$ can be extended continuously to a plurisubharmonic function on $\Omega$. The aim of this paper is to study an analogue of this concept in the category of unbounded domains in $C^n$. The use of Jensen measures relative to classes of plurisubharmonic functions plays a key role in our work

We give a sufficient condition for a $C^{\omega }$ (resp. $C^{\infty }$)-totally real, complex-tangential, $(n-1)$-dimensional submanifold in a weakly pseudoconvex boundary of class $C^{\omega }$ (resp. $C^{\infty }$) to be a local peak set for the class $𝒪$ (resp. $A^{\infty }$). Moreover, we give a consequence of it for Catlin’s multitype.

Let $D\subset {ℂ}^n$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_D(.,.)$. In this article we give for a compact subset $K\subset D$ a quantitative upper bound for the supremum ${sup}_{z\in K}\left|G_D(z,w)\right|$ in terms of the boundary distance of $K$ and $w$. This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_D(w;X)$ tends to infinity, for $X\in {ℂ}^n/\left\{O\right\}$, when $w\in D$ tends to a boundary point....