Abstract:

Let K be a compact plane set and let Ω = C* \ K, where C* is the Riemann sphere. This talk concerns the problem of determining when there exist non-constant bounded analytic functions on Ω. Several necessary, sufficient, or necessary and sufficient conditions will be discribed, some very old and some very new. I will try to explain how this problem uses ideas from Potential Theory, Harmonic Analysis, Geometric Measure Theory and Functional Analysis.