Narrow Results

We prove that every locally pluripolar set on a compact complex manifold is pluripolar. This extends similar results in the Kähler case.

Let $\phi$ be a function in the complex Sobolev space $W^{\ast }(U)$, where $U$ is an open subset in ${ℂ}^k$. We show that the complement of the set of Lebesgue points of $\phi$ is pluripolar. The key ingredient in our approach is to show that $|\phi {|}^{\alpha }$ for $\alpha \in [1,2)$ is locally bounded from above by a plurisubharmonic function.

The aim of paper is to find the condition under which a Fréchet-valued function $f\in C^{\infty }(\{0\})$ admitting meromorphic extension along some pencil of complex lines can be meromorphically extended to a neighborhood of $0\in {ℂ}^N.$ Some auxiliary results concerning the domains of existence for Fréchet-valued meromorphic functions, Rothstein’s theorem, Levi extension theorem for meromorphic functions with values in a locally complete space, convergence of formal power series of Fréchet-valued homogeneous polynomials are also proved in this work.