Search results for "Analytic capacity"

Rectifiability and analytic capacity in the complex plane

1995

Analytic capacity and removable sets In this chapter we shall discuss a classical problem in complex analysis and its relations to the rectifiability of sets in the complex plane C . The problem is the following: which compact sets E ⊃ C are removable for bounded analytic functions in the following sense? (19.1) If U is an open set in C containing E and f : U\E → C is a bounded analytic function, then f has an analytic extension to U . This problem has been studied for almost a century, but a geometric characterization of such removable sets is still lacking. We shall prove some partial results and discuss some other results and conjectures. For many different function classes a complete so…

Weakly compact composition operators between algebras of bounded analytic functions

1999

Analytic capacity and quasiconformal mappings with $W^{1,2}$ Beltrami coefficient

2008

We show that if $\phi$ is a quasiconformal mapping with compactly supported Beltrami coefficient in the Sobolev space $W^{1,2}$, then $\phi$ preserves sets with vanishing analytic capacity. It then follows that a compact set $E$ is removable for bounded analytic functions if and only if it is removable for bounded quasiregular mappings with compactly supported Beltrami coefficient in $W^{1,2}$.

Singular integrals, analytic capacity and rectifiability

1997

In this survey we study some interplay between classical complex analysis (removable sets for bounded analytic functions), harmonic analysis (singular integrals), and geometric measure theory (rectifiability).

Lipschitz classes and the Hardy-Littlewood property

1993

We study the geometry of plane domains and the uniform Holder continuity properties of analytic functions.