Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains
Annales de l'Institut Fourier, Volume 28 (1978) no. 4, pp. 147-167.

On a Lipschitz domain D in R n , three theorems on harmonic functions are proved. The first (boundary Harnack principle) compares two positive harmonic functions at interior points near an open subset of the boundary where both functions vanish. The second extends some familiar geometric facts about the Poisson kernel on a sphere to the Poisson kernel on D. The third theorem, on non-tangential limits of quotient of two positive harmonic functions in D, generalizes Doob’s relative Fatou theorem on a sphere. The main tools are maximum principle, Harnack inequality and differentiation of measures.

Nous obtenons trois théorèmes sur les fonctions harmoniques dans un domaine lipschitzien : un principe du type de Harnack sur la frontière ; des inégalités géométriques pour le noyau de Poisson d’un tel domaine ; un théorème relatif de Fatou. Les outils essentiels sont le principe du maximum, l’inégalité de Harnack, et la dérivation des mesures.

@article{AIF_1978__28_4_147_0,
     author = {Wu, Jang-Mei G.},
     title = {Comparisons of kernel functions boundary {Harnack} principle and relative {Fatou} theorem on {Lipschitz} domains},
     journal = {Annales de l'Institut Fourier},
     pages = {147--167},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {28},
     number = {4},
     year = {1978},
     doi = {10.5802/aif.719},
     zbl = {0368.31006},
     mrnumber = {80g:31005},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.719/}
}
TY  - JOUR
TI  - Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains
JO  - Annales de l'Institut Fourier
PY  - 1978
DA  - 1978///
SP  - 147
EP  - 167
VL  - 28
IS  - 4
PB  - Imprimerie Durand
PP  - 28 - Luisant
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.719/
UR  - https://zbmath.org/?q=an%3A0368.31006
UR  - https://www.ams.org/mathscinet-getitem?mr=80g:31005
UR  - https://doi.org/10.5802/aif.719
DO  - 10.5802/aif.719
LA  - en
ID  - AIF_1978__28_4_147_0
ER  - 
%0 Journal Article
%T Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains
%J Annales de l'Institut Fourier
%D 1978
%P 147-167
%V 28
%N 4
%I Imprimerie Durand
%C 28 - Luisant
%U https://doi.org/10.5802/aif.719
%R 10.5802/aif.719
%G en
%F AIF_1978__28_4_147_0
Wu, Jang-Mei G. Comparisons of kernel functions boundary Harnack principle and relative Fatou theorem on Lipschitz domains. Annales de l'Institut Fourier, Volume 28 (1978) no. 4, pp. 147-167. doi : 10.5802/aif.719. https://aif.centre-mersenne.org/articles/10.5802/aif.719/

[1] A. Ancona, Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien, Ann. Inst. Fourier, (to appear). | EuDML: 74379 | Numdam | Zbl: 0377.31001

[2] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions II, Proc. Cambridge Philos. Soc., 42 (1946), 1-10. | MR: 7,281e | Zbl: 0063.00353

[3] M. Brelot, Remarques sur les zéros à la frontière des fonctions harmoniques positives, Un. Mat. Ita., Boll., Suppl., Ser. 4, 12 (1975), 314-319. | MR: 54 #7823 | Zbl: 0338.31004

[4] M. Brelot et J. L. Doob, Limites angulaires et limites fines, Ann. Institut Fourier, 13, 2 (1963), 395-415. | EuDML: 73813 | Numdam | MR: 33 #4299 | Zbl: 0132.33902

[5] B. Dahlberg, On estimates of harmonic measure, Arch. Rational Mech. Anal., 65, N° 3 (1977), 275-288. | MR: 57 #6470 | Zbl: 0406.28009

[6] J. L. Doob, A relativized Fatou theorem, Proc. Nat. Acad. Sc., 45 (1959), N° 2, 215-222. | MR: 21 #5822 | Zbl: 0106.07801

[7] K. Gowrisankaran, Fatou-Naim-Doob limit theorems in the axiomatic system of Brelot, Ann. Inst. Fourier, 16, 2 (1966), 455-467. | EuDML: 73911 | Numdam | MR: 35 #1802 | Zbl: 0145.15103

[8] R. A. Hunt and R. L. Wheeden, On the boundary values of harmonic functions, Trans. Amer. Math. Soc., 132 (1968), 307-322. | MR: 37 #1634 | Zbl: 0159.40501

[9] R. A. Hunt and R. L. Wheeden, Positive harmonic functions on Lipschitz domains, Trans. Amer. Math. Soc., 147 (1970), 507-527. | MR: 43 #547 | Zbl: 0193.39601

[10] J. T. Kemper, A boundary Harnack principle for Lipschitz domains and the principle of positive singularities, Comm. Pure Applied Math., 25 (1972), 247-255. | MR: 45 #2193 | Zbl: 0226.31007

[11] J.-M. Wu, On functions subharmonic in a Lipschitz domain, Proc. Amer. Math. Soc. (to appear). | Zbl: 0377.31007

Cited by Sources: