Estimates of harmonic measures
Lennart Carleson · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1982
CARLESONl.The theory of analytic functions of one complex variable can be based on essentially three different principles: (l) the Cauchy integral and the related power series expansions, (2) the geometric idea of conformal mapping and (3) the use of harmonic functions and the study of loglf(z)1.In the development during the last fifty years the last aspect has been predominant and the work of R. Nevanlinna together with that of many other Scandinavian mathematicians has been of funda- mental importance.To combine the aspects (2) and ( 3) one must be able to handle harmonic functions in terms of geometric conditions and this is how harmonic measures enter.I shall here give a short summary of the most important methods and mention some open probiems.During the last decades the first method has been revived through the D-equation and we now have methods where all three aspects can be made to work together.Assume that I is a domain in the plane.We divide its boundary into two sub- sets, 09:Eu.l-,and we are interested in estimating the harmonic function in 9, co(z;E;9) which :1 on E and :0 on J-, at some fixed interior point zoof 9.This point zo is always assumed not to be close to 09.I shall schematically classify the situation in the following sections. I simply connected; E an arcThis case is the most well-known and there are three essentially equivalent methods (A) the Ahlfors distortion theorem [1] (B) Beurling's method of extremal length (C) Carleman's differential inequality [4].Since Beurling's method is the least known and the most flexible, I shall indicate the result.We assume that P is some fixed arc "close" to zo and that the geometry of I is well-behaved in this part of 9. Consider the family of all curves {y} joining B