The Carleson measure and meromorphic functions of uniformly bounded characteristic

Some Unsolved Problems on Meromorphic Functions of Uniformly Bounded Characteristic

The family UBC(R) of meromorphic functions of uniformly bounded characteristic in a Rieman surface R is defined in terms of the Shimizu-Ahlfors characteristic function. There are some natural parallels between UBC(R) and BMOA(R), the fsmily of holomorr.hc f11nctonz of bounded mesn oscilltion in R. After a survey some open problems are proposed in contrast with BMOA(R).

On Meromorphic Functions of Bounded Characteristic

Functions of Uniformly Bounded Characteristic on Riemann Surfaces

A characteristic function T(D, w, f) of Shimizu and Ahlfors type for a function / meromorphic in a Riemann surface R is defined, where D is a regular subdomain of R containing a reference point w G R. Next we suppose that R has the Green functions. Letting T(w,f) = Mthdir T{D,w,f), we define / to be of uniformly bounded characteristic in R, f G UBC(i?) in notation, if sup^g/j T(w, f) < oo. We s...

Valuations on Meromorphic Functions of Bounded Type

The primary purpose of this paper is to show that every valuation on the field of meromorphic functions of bounded type on a finitely sheeted unlimited covering Riemann surface is a point valuation if and only if the same is true on its base Riemann surface. The result is then applied to concrete examples and some related results are obtained. Any valuation on the field M(W) of single valued me...

Uniform Rectifiability, Carleson Measure Estimates, and Approximation of Harmonic Functions

Let E ⊂ Rn+1, n ≥ 2, be a uniformly rectifiable set of dimension n. Then bounded harmonic functions in Ω := Rn+1 \ E satisfy Carleson measure estimates, and are “ε-approximable”. Our results may be viewed as generalized versions of the classical F. and M. Riesz theorem, since the estimates that we prove are equivalent, in more topologically friendly settings, to quantitative mutual absolute con...