Abstract
There are two parts in this thesis. In the first part, let D contained in $C^n$, n>= 2, be a smooth bounded pseudoconvex domain and K(z,w) be the Bergman kernel function associated with D. We show that, if the points of infinite type are isolated on the boundary bD, then K(z,w) belongs to $C^\infty(\overline D\times\overline D\setminus\Delta(bD))$, where $\Delta(bD)$={(z,z):z in bD}. Secondly, assume that D is convex Reinhardt and that K(z,w) belongs to $C^\infty(\overline D\times\overline D\setminus\Delta(bD))$, we prove that bD satisfy property (P),i.e., there exists a family of uniformly bounded plurisubharmonic functions that are smooth up to bD and with arbitrarily large Hessians on the boundary. Thirdly, exploiting fine topology, we exhibit explicitly a domain D without analytic disc on bD and nonetheless does not possess property (P). In the second part, consider a bounded domain D in $C^n$, n>= 2, with real analytic boundary bD. Assume that either D is pseudoconvex or D satisfies condition R, i.e., the orthogonal projection P from $L^2(D)$ to the space of square integrable holomorphic functions on D, maps $C^\infty(\overline D)$ continuously into itself. Let Aut(D) be the group of biholomorphic mappings from D onto D equipped with the compact-open topology. Then for each g in Aut(D) one can find a neighborhood $V_g$ of g in Aut(D) and a neighborhood $\Omega_g$ of $\overline D$ such that every f in $V_g$ admits a holomorphic extension to $\Omega_g$ and moreover, the action $(f,z)\mapsto f(z)$ is real analytic with respect to the joint variable (f,z) on $V_g\times\Omega_g$. Assume further that Aut(D) is compact. Then there exists a fixed neighborhood $\Omega$ of $\overline D$ such that every f in Aut(D) can be extended holomorphically to $\Omega$ and the action $(f,z)\mapsto f(z)$ is real analytic on Aut}(D) $\times\Omega$.