International Congress of Mathematicians

704 C. E. Kenigvariables) and, by Preiss's result, OJ°° is asymptotically optimal doubling, but, ofcourse, 0 is not Reifenberg vanishing, since it is -A^-Reifenberg flat, and no better.Our converse to Theorem 1 is now:Theorem 2. ([16]) Assume that O C R" +1 is an NTA, and thatoj (OJ°°) is asymptoticallyoptimal doubling. Ifn= 1,2, then 0 is Reifenberg vanishing. Ifn>3 and0 is ö-Reifenberg flat, ö < -fi^, then 0 is Reifenberg vanishing.This is in fact a GAIT result. It remains valid if OJ (OJ°°) is replaced by anyasymptotically optimal doubling measure p with support 90. The idea of the proofis to use a "blow-up" argument to reduce matters to the Kowalski-Preiss theorem.Further GAIT results along these lines, also in the higher codimension case, wereobtained by David-Kenig-Toro [4].We now turn to the results motivated by (II)o- Note that the unit normaln satisfies \n\ = 1, and so the BAIO condition on it is automatic, but the VAIOcondition is not. To put our work in perspective, we recall some of the history ofthe subject.A domain 0 C R 1+1 = R 2 is called a chord-arc domain is 90 is locally rectifiable,and, whenever Qi,Q2 £ 90, we have £(s(Qi,Q2)) < C\Qi — Q2I, where £denotes length and s(Qi,Qfi) is the shortest arc between Qi and Q2- 0 is calledvanishing chord-arc if, in addition, as Qi —^ Q2, the ratio,A AA tends to 1, uni-\W 1 W2 [formly on compact sets. The first person to study harmonic measure on chord-arcdomains in the plane was Lavrentiev ([18]), who proved:Theorem. ([18]) //OC R 1+1 is chord-arc, then doj = kda with logfc £ BAlO(da).(In fact, OJ £ A 00 (da), the Muckenhoupt class [7].)For vanishing chord-arc domains in the plane, Pommerenke [19] proved:Theorem. ([19]) Suppose that ii is a chord-arc domain in R 1+1 . Then 0 is vanishingchord-arc if and only if doj = kda with logfc £ VAlO(da).These results were obtained using function theory, so their proofs don't generalizeto higher dimensions. In higher dimensions, the first breakthrough came in thecelebrated theorem of B. Dahlberg [2], who showed that, if 0 C R" +1 is a Lipschitzdomain, then doj = kda with logfc £ BAIO (in fact, OJ £ A 00 (daj). One directionof Pommerenke's result was extended to higher dimensions by Jerison-Kenig [10],who showed that, if 0 is a C 1 domain, then log k £ VAIO. (In general, note that0 is of class C 1 need not imply that log k is continuous.) In order to explain ourresults and to clarify the connection with condition (II)o, we need to introduce someterminology. A domain 0 C R" +1 will be calld a chord-arc domain if it is an NTAdomain (see [9]) of locally finite perimeter (see [6]) and its boundary is Ahlfors regular,i.e., the surface measure a (which is Radon measure on 90, by the assumptionof locally finite perimeter) satisfies the inequalitiesC- X r n < a(B(r, Q) n 90) < Cr n(for Q ë An 90, K CC R" +1 and small r; or, if 0 is an unbounded NTA, forall Q £ 90 and r > 0). A fundamental result of David-Jerison [3] and Semmes