No abstract uploaded!

Let$M$be a smooth hypersurface of constant signature in$CP$^{$n$},$n$≥3. We prove the regularity foron$M$in bidegree (0,1). As a consequence, we show that there exists no smooth hypersurface in$CP$^{$n$},$n$≥3, whose Levi form has at least two zero-eigenvalues.

We consider the convolution operators in spaces of functions which are holomorphic in a bounded convex domain in ℂ^{$n$}and have a polynomial growth near its boundary. A characterization of the surjectivity of such operators on the class of all domains is given in terms of low bounds of the Laplace transformation of analytic functionals defining the operators.

We show that a family $\mathcal{F}$ of analytic functions in the unit disk ${\mathbb{D}}$ all of whose zeros have multiplicity at least$k$and which satisfy a condition of the form $$f^n(z)f^{(k)}(xz)\ne1$$ for all $z\in{\mathbb{D}}$ and $f\in\mathcal{F}$ (where$n$≥3,$k$≥1 and 0<|$x$|≤1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman–Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.

We show how to construct the Perron–Bremermann function by using proper analytic disks. We apply this result to the polynomial hull of a compact set$K$defined on the boundary of the unit ball.