Doubling conditions for harmonic measure in John domains  [ Condition de doublement pour la mesure harmonique dans les domaines de John ]
Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 429-445.

Nous introduisons des classes nouvelles de domaines, domaines semi-uniformes et domaines intérieurs semi-uniformes. Elles sont intermédiaires entre la classe des domaines de John et la classe des domaines uniformes. Sous la condition de densité de capacité, nous prouvons que la mesure harmonique d’un domaine D de John satisfait certaines conditions de doublement si et seulement si D est un domaine semi-uniforme ou un domaine intérieur semi-uniforme.

We introduce new classes of domains, i.e., semi-uniform domains and inner semi-uniform domains. Both of them are intermediate between the class of John domains and the class of uniform domains. Under the capacity density condition, we show that the harmonic measure of a John domain D satisfies certain doubling conditions if and only if D is a semi-uniform domain or an inner semi-uniform domain.

DOI : https://doi.org/10.5802/aif.2357
Classification : 31B05,  31B25,  31C35
Mots clés : John domain, semi-uniform domain, inner semi-uniform domain, harmonic measure, doubling condition, capacity density condition
@article{AIF_2008__58_2_429_0,
     author = {Aikawa, Hiroaki and Hirata, Kentaro},
     title = {Doubling conditions for harmonic measure in John domains},
     journal = {Annales de l'Institut Fourier},
     pages = {429--445},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     doi = {10.5802/aif.2357},
     mrnumber = {2410379},
     zbl = {1151.31004},
     language = {en},
     url = {www.numdam.org/item/AIF_2008__58_2_429_0/}
}
Aikawa, Hiroaki; Hirata, Kentaro. Doubling conditions for harmonic measure in John domains. Annales de l'Institut Fourier, Tome 58 (2008) no. 2, pp. 429-445. doi : 10.5802/aif.2357. http://www.numdam.org/item/AIF_2008__58_2_429_0/

[1] Aikawa, H. Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan, Volume 53 (2001) no. 1, pp. 119-145 | Article | MR 1800526 | Zbl 0976.31002

[2] Aikawa, H. Hölder continuity of the Dirichlet solution for a general domain, Bull. London Math. Soc., Volume 34 (2002) no. 6, pp. 691-702 | Article | MR 1924196 | Zbl 1036.31003

[3] Aikawa, H.; Hirata, K.; Lundh, T. Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan, Volume 58 (2006) no. 1, pp. 247-274 | Article | MR 2204573 | Zbl 1092.31006

[4] Aikawa, H.; Lundh, T.; Mizutani, T. Martin boundary of a fractal domain, Potential Anal., Volume 18 (2003) no. 4, pp. 311-357 | Article | MR 1953266 | Zbl 1021.31002

[5] Ancona, A. On strong barriers and an inequality of Hardy for domains in R n , J. London Math. Soc. (2), Volume 34 (1986) no. 2, pp. 274-290 | Article | MR 856511 | Zbl 0629.31002

[6] Balogh, Z.; Volberg, A. Boundary Harnack principle for separated semihyperbolic repellers, harmonic measure applications, Rev. Mat. Iberoamericana, Volume 12 (1996) no. 2, pp. 299-336 | MR 1402670 | Zbl 0857.31006

[7] Balogh, Z.; Volberg, A. Geometric localization, uniformly John property and separated semihyperbolic dynamics, Ark. Mat., Volume 34 (1996) no. 1, pp. 21-49 | Article | MR 1396621 | Zbl 0855.30022

[8] Bass, R. F.; Burdzy, K. A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2), Volume 134 (1991) no. 2, pp. 253-276 | Article | MR 1127476 | Zbl 0747.31008

[9] Bonk, M.; Heinonen, J.; Koskela, P. Uniformizing Gromov hyperbolic spaces, Astérisque, Volume 270 (2001), pp. viii+99 | MR 1829896 | Zbl 0970.30010

[10] Gehring, F. W.; Martio, O. Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., Volume 10 (1985), pp. 203-219 | MR 802481 | Zbl 0584.30018

[11] Gehring, F. W.; Martio, O. Quasiextremal distance domains and extension of quasiconformal mappings, J. Analyse Math., Volume 45 (1985), pp. 181-206 | Article | MR 833411 | Zbl 0596.30031

[12] Gehring, F. W.; Osgood, B. G. Uniform domains and the quasihyperbolic metric, J. Analyse Math., Volume 36 (1979), pp. 50-74 | Article | MR 581801 | Zbl 0449.30012

[13] Jerison, D. S.; Kenig, C. E. Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math., Volume 46 (1982) no. 1, pp. 80-147 | Article | MR 676988 | Zbl 0514.31003

[14] Kim, K.; Langmeyer, N. Harmonic measure and hyperbolic distance in John disks, Math. Scand., Volume 83 (1998) no. 2, pp. 283-299 | MR 1673938 | Zbl 0927.30015

[15] Väisälä, J. Uniform domains, Tohoku Math. J. (2), Volume 40 (1988) no. 1, pp. 101-118 | Article | MR 927080 | Zbl 0627.30017

[16] Väisälä, J. Relatively and inner uniform domains, Conform. Geom. Dyn., Volume 2 (1998), pp. 56-88 ((electronic)) | Article | MR 1637079 | Zbl 0902.30017