Exceptional sets for holomorphic Sobolev functions.

Singular sets of Sobolev functions

Exceptional Sets for Universally Polygonally Approximable Functions

In [9], the present authors and Richard O’Malley showed that in order for a function be universally polygonally approximable it is necessary that for each ε > 0, the set of points of non-quasicontinuity be σ − (1− ε) symmetrically porous. The question as to whether that condition is sufficient or not was left open. Here we prove that if a set, E = S∞ n=1 En, such that each Ei is closed and 1-sy...

Exceptional values in holomorphic families of entire functions

For a holomorphic family of entire functions we study dependence of Picard exceptional values of these functions on the parameter. In 1926, Julia [5] studied singularities of implicit functions defined by equations f(z, w) = 0, where f is an entire function of two variables such that f(., w) 6≡ 0 for every w ∈ C. Among other things, he investigated the exceptional set P consisting of those w fo...

Maximal cluster sets on spaces of holomorphic functions

In this paper, which has an expository nature, we consider the so called cluster sets, that is the set of accumulation points of the set of values taken by a (usually, holomorphic) function on prescribed subsets. We are specially interested in the action of holomorphic operators in order to obtain maximal cluster sets, see below for details. Extensions of classical cluster sets, where they are ...

Zero sets of holomorphic functions in the bidisc

In this work we characterize the zero sets of holomorphic functions f in the bidisc such that log jf j Lp D p Moreover we give a su cient condition on a analytic variety to be de ned by a function in A D Introduction In this paper we study some geometrical conditions on analytic varieties in the bidisc D fz C jz j jz j g to be de ned by an holomorphic func tion with some restriction on its grow...