Doubling conditions for harmonic measure in John domains
[Condition de doublement pour la mesure harmonique dans les domaines de John]
Aikawa, Hiroaki1 ; Hirata, Kentaro2
1 Hokkaido University Department of Mathematics Sapporo 060-0810 (Japan)
2 Akita University Faculty of Education and Human Studies Akita 010-8502 (Japan)
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 : 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
Aikawa, Hiroaki 1 ; Hirata, Kentaro 2

1 Hokkaido University Department of Mathematics Sapporo 060-0810 (Japan)
2 Akita University Faculty of Education and Human Studies Akita 010-8502 (Japan) 
@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{\textquoteright}institut Fourier},
     volume = {58},
     number = {2},
     year = {2008},
     doi = {10.5802/aif.2357},
     zbl = {1151.31004},
     mrnumber = {2410379},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2357/}
}
TY  - JOUR
AU  - Aikawa, Hiroaki
AU  - Hirata, Kentaro
TI  - Doubling conditions for harmonic measure in John domains
JO  - Annales de l'Institut Fourier
PY  - 2008
SP  - 429
EP  - 445
VL  - 58
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2357/
DO  - 10.5802/aif.2357
LA  - en
ID  - AIF_2008__58_2_429_0
ER  - 
%0 Journal Article
%A Aikawa, Hiroaki
%A Hirata, Kentaro
%T Doubling conditions for harmonic measure in John domains
%J Annales de l'Institut Fourier
%D 2008
%P 429-445
%V 58
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2357/
%R 10.5802/aif.2357
%G en
%F 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://archive.numdam.org/articles/10.5802/aif.2357/

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

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

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

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

Cité par Sources :