International Congress of Mathematicians

708 C. E. KenigSince log/i G VA10(90), the average av B^.^hda is close to the value of log/iin a proportionally large subset of B(r, Q) n 90. This remark allows us to concludethat (7.6) holds, which is crucial to the application and which fails in general underjust (7.1) and (7.2).As an immediate application of Theorems 6 and 7, we obtain that Ooo is ahalf-plane. This already establishes that 0 is Reifenberg vanishing in Theorem 5.To establish that n is in VAIO, we assume otherwise, and obtain Q t —t Q œ , r t —t 0,such that av B (ri,Qi)\n — nB( ri ,Qi)\ 2 da > £ 2 , £ > 0. We consider the correspondingblow-up sequence, and let e n +i be the direction perpendicular to 90^. By thedivergence theorem and (7.1) and (7.2), we have for p £ Cg°(W n+1 ) thatand hencelim / p(Hi,e n+ i)dai = / pdx'Ofi;Jll"x{0}2—»OOso that (7.6) yieldslim -j / pdai - - j Lp\rti — e n+ i\ 2 dai > = / pdx,l—»oo Ofi; 2 J dQ . J jR»x{0}lim / p\n,i — e n+ iYdai = 0.2—»OOTaking p > XB(I,Q) yields the corresponding bound for the integral on 90j C\B(1,0).Butand henceav B(l,0)nOfiil n * — e »+l|aa i — ay B(r i ,Qi)\ n — e n +i| OCT,, xl/2 , xl/2£< lim (av B(r Qi) |n^n B ( r Q;)! 2^) < 2 lim (av B(r ^cfin - e„+i| 2 do-) ,i—»oo \ / i—»oo \ /a contradiction. This concludes the proof.References[1] H. W. Alt and L. A. Caffarelli, Existence and regularity for a minimum problemwith free boundary, J. Reine Angew. Math., 325 (1981), 105^144.[2] B. Dahlberg, On estimates for harmonic measure, Arch. Rat. Mech. Anal., 65(1977), 272^288.[3] G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonicmeasure and singular integrals, Indiana Univ. Math. J., 39 (1990), 831A345.[4] G. David, C. Kenig, and T. Toro, Asymptotically optimally doubling measuresand Reifenberg flat sets with vanishing constant, CPAM, 54 (2001), 385^449.[5] P. Düren, The theory of H p spaces, Academic Press, New York, 1970.[6] L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions,Studies in Advanced Alathematics, CRC Press, 1992.[7] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities andrelated topics, Alath. Studies, no. 116, North Holland, 1985.