On the boundary limits of harmonic functions with gradient in L p
Annales de l'Institut Fourier, Tome 34 (1984) no. 1, pp. 99-109.

Dans cet article on étudie l’allure tangentielle à la frontière des fonctions harmoniques dans la classe de Sobolev W 1 p (R + n ), où R + n est le demi-espace de R n . On donne une généralisation du résultat récent de Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), dans le cas p=n. Ici on utilise la représentation intégrale des fonctions de Beppo-Levi de Ohtsuka (Lecture Notes, Hiroshima Univ., 1973), et notre méthode est différente de celle de Nagel, Rudin et Shapiro (Ann. of Math., 116 (1982), 331–360).

This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in L n (R + n ), R + n denoting the upper half space of the n-dimensional euclidean space R n . Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation of precise functions given by Ohtsuka (Lecture Notes, Hiroshima Univ., 1973).

@article{AIF_1984__34_1_99_0,
     author = {Mizuta, Yoshihiro},
     title = {On the boundary limits of harmonic functions with gradient in $L^p$},
     journal = {Annales de l'Institut Fourier},
     pages = {99--109},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     number = {1},
     year = {1984},
     doi = {10.5802/aif.952},
     mrnumber = {85f:31009},
     zbl = {0522.31009},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_1984__34_1_99_0/}
}
Mizuta, Yoshihiro. On the boundary limits of harmonic functions with gradient in $L^p$. Annales de l'Institut Fourier, Tome 34 (1984) no. 1, pp. 99-109. doi : 10.5802/aif.952. https://aif.centre-mersenne.org/item/AIF_1984__34_1_99_0/

[1] L. Carleson, Selected Problems on exceptional sets, Van Nostrand, Princeton, 1967. | MR 37 #1576 | Zbl 0189.10903

[2] A.B. Cruzeiro, Convergence au bord pour les fonctions harmoniques dans Rd de la classe de Sobolev Wd1, C.R.A.S., Paris, 294 (1982), 71-74. | MR 83g:31006 | Zbl 0495.31003

[3] N.G. Meyers, A theory of capacities for potentials in Lebesgue classes, Math. Scand., 26 (1970), 255-292. | MR 43 #3474 | Zbl 0242.31006

[4] N.G. Meyers, Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR 51 #3477 | Zbl 0334.31004

[5] Y. Mizuta, On the existence of boundary values of Beppo Levi functions defined in the upper half space of Rn, Hiroshima Math. J., 6 (1976), 61-72. | MR 56 #8878 | Zbl 0329.31007

[6] Y. Mizuta, Existence of various boundary limits of Beppo Levi functions of higher order, Hiroshima Math. J., 9 (1979), 717-745. | MR 81d:31013 | Zbl 0475.31004

[7] A. Nagel, W. Rudin and J.H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math., 116 (1982), 331-360. | MR 84a:31002 | Zbl 0531.31007

[8] M. Ohtsuka, Extremal length and precise functions in 3-space, Lecture Notes, Hiroshima Univ., 1973.

[9] E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, 1970. | MR 44 #7280 | Zbl 0207.13501

[10] H. Wallin, On the existence of boundary values of a class of Beppo Levi functions, Trans. Amer. Math. Soc., 120 (1965), 510-525. | MR 32 #5911 | Zbl 0139.06301

[11] J.-M. G. Wu, Lp-densities and boundary behaviors of Green potentials, Indiana Univ. Math. J., 28 (1979), 895-911. | Zbl 0449.31003

[12] W.P. Ziemer, Extremal length as a capacity, Michigan Math. J., 17 (1970), 117-128. | MR 42 #3299 | Zbl 0183.39104