International Congress of Mathematicians

Harmonie Aleasure and "Locally Flat" Domains 707Suppose that sup x€fi |Vu(x)| < 1 and h(Q) > 1 for (da-)a.ca half-space and u(x,x n+ i) = x n+ i.Q on dil. Then 0 isThis allows us to prove the crucial blow-up result, which we now describe.Let 0 be as in Theorem 5, and assume in addition that 0 is unbounded. Supposedoj°° = h da with log h £ VMO(da), and let u be the associated harmonic function.Let Qi £ 90 and assume that Qi —¥ Qoo £ 90 as i —¥ oo (without loss of generality,Qoo = 0). Let {r,}^1 be a sequence of positive numbers tending to 0, and putiii1 (0 - Qi düi1 (90 - QiUifiX)1u(r t X + Qi) and doj°° = hi(Q)dai(Q),where h t (Q) = aVB(r. ^ft^M^Q + Qi)- Then:Theorem 7. There exists a subsequence of {Oj} (which we will call again {Hi})satisfying:andOj —t OQO in the Hausdorff distance sense, uniformly on compact sets; (7.1)90j —t diioo in the Hausdorff distance sense, uniformly on compact sets; (7.2)«« —^ «oo uniformly on compact sets (7.3)AUQO = 0 in OQO«oo > 0 in ii r«no =0 in diirx(7.4)Furthermore(7.5)ando-i ->• CT«», (7.6)weakly as Radon measures. Here, CTQO = / ri n [dii^ and WQO denotes the harmonicmeasure of iloo with pole at oo (corresponding to «ooj- Moreover,SUp |V«oo(^)| < 12Gfioo(7.7)andhoo(Q)dojr,da r . '-(Q)>1 forHn -a.c Q £ 90 œ . (7.8)