Harmonic Measure and Estimates of Green's Function

The study of exceptional sets for nontangential limits of harmonic functions has led to the problem of the equivalence of harmonic measure and surface measure on the boundary of Lipschitz domains. In this note we will review the development of these results. We will see how estimates of Green's function can be used to relate the measures and look at methods of obtaining the desired estimates. If u is positive and harmonic in the unit ball B c En+\9 the Poisson integral representation leads easily to finite nontangential limits at almost every point of 32?. We are interested in modifications and generalizations of this result In 1950, A. P. Calderón [2] proved the following: If u is harmonic in the unit ball B c En+x and for each point Q E E, E c dB, u is bounded in some truncated cone T(Q) with vertex at Q, then u has a finite nontangential limit at almost every Q E E. In Calderón's result, we may assume the cones T(Q) axe tangent to, and truncated by the more distant surface of, a fixed small ball with center at the origin. In that case, B = UQBE^(Q) i s a starlike Lipschitz domain. Calderón's result could be stated with the hypothesis that u is harmonic and bounded in B. In 1962, L. Carleson [4] obtained the same conclusion as in Calderón's result, but with the hypothesis that u is harmonic and bounded from below in Ê. In his proof, Carleson introduced the harmonic measure associated with the domain Ê. An estimate of Green's function for B was used to obtain results in terms of surface measure. In 1964, K.-0. Widman [11] showed if u is harmonic and positive in a domain D with 35 G C , e > 0, then u has finite nontangential limits at almost every point of dD. As in the proof of Carleson, harmonic measure was used and an estimate of Green's function showed the exceptional set was of zero surface measure. Following the general outline of Carleson's proof, Hunt and Wheeden [8] in 1970 proved that functions which are positive and harmonic in domains D with dD E Lip(l) have finite nontangential limits except on a set of harmonic measure zero. The fact that this exceptional set is also of surface measure zero is a consequence of the 1976 result of B. Dahlberg [6] that harmonic measure and surface measure are equivalent on the boundary of Lipschitz domains. Let us briefly review a method to obtain the nontangential limits. We assume that D c En+l9 n > 2, 3D E Lip(l) and D is starlike about a point PQ. In particular, we assume there is a fixed truncated open cone F which is contained in D whenever it is positioned with vertex at Q E dD and axis