International Congress of Mathematicians

702 C. E. Kenigin Q with u| = 0, we define doj°°, harmonic measure with pole at infinity, to bethe measure satisfyingpdw 00 = [ vAip, for p £ C 0 °°(O).an JnThe existence and uniqueness of v and OJ 00 (modulo multiplicative constants) canbe established, for instance, when Q is an unbounded NTA (non-tangentially accessible)domain abbreviation is clarified later, it might as well be here, (see [16] fordetails). For example, if 0 = R" +1 = {(x,t) : t > 0}, then v(x,t) = t and du>°° = dxon R". The work I will describe originated from trying to understand, as a --I 0,the classical theorem of Kellogg, which shows that, if 0 is of class C 1,a , 0 < a < 1,then dw = kda with log k £ C a ; and its "converse", the free boundary regularity ofAlt-Caffarelli [1], which states that, if 0 satisfies certain necessary weak conditions(to be more fully explained later) and du = kda with logfc £ C a , then Q must beof class C 1,a .To motivate our results, we recall real variable characterizations of C 1,a andC a :p £ C 1 ' a (W l )(0