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 , denoting the upper half space of the -dimensional euclidean space . 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).
Dans cet article on étudie l’allure tangentielle à la frontière des fonctions harmoniques dans la classe de Sobolev , où est le demi-espace de . On donne une généralisation du résultat récent de Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), dans le cas . 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).
@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 = {Imprimerie Louis-Jean}, address = {Gap}, volume = {34}, number = {1}, year = {1984}, doi = {10.5802/aif.952}, mrnumber = {85f:31009}, zbl = {0522.31009}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.952/} }
TY - JOUR AU - Mizuta, Yoshihiro TI - On the boundary limits of harmonic functions with gradient in $L^p$ JO - Annales de l'Institut Fourier PY - 1984 SP - 99 EP - 109 VL - 34 IS - 1 PB - Imprimerie Louis-Jean PP - Gap UR - http://archive.numdam.org/articles/10.5802/aif.952/ DO - 10.5802/aif.952 LA - en ID - AIF_1984__34_1_99_0 ER -
%0 Journal Article %A Mizuta, Yoshihiro %T On the boundary limits of harmonic functions with gradient in $L^p$ %J Annales de l'Institut Fourier %D 1984 %P 99-109 %V 34 %N 1 %I Imprimerie Louis-Jean %C Gap %U http://archive.numdam.org/articles/10.5802/aif.952/ %R 10.5802/aif.952 %G en %F 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, Volume 34 (1984) no. 1, pp. 99-109. doi : 10.5802/aif.952. http://archive.numdam.org/articles/10.5802/aif.952/
[1] Selected Problems on exceptional sets, Van Nostrand, Princeton, 1967. | MR | Zbl
,[2] 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 | Zbl
,[3] A theory of capacities for potentials in Lebesgue classes, Math. Scand., 26 (1970), 255-292. | MR | Zbl
,[4] Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR | Zbl
,[5] 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 | Zbl
,[6] Existence of various boundary limits of Beppo Levi functions of higher order, Hiroshima Math. J., 9 (1979), 717-745. | MR | Zbl
,[7] Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math., 116 (1982), 331-360. | MR | Zbl
, and ,[8] Extremal length and precise functions in 3-space, Lecture Notes, Hiroshima Univ., 1973.
,[9] Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, 1970. | MR | Zbl
,[10] On the existence of boundary values of a class of Beppo Levi functions, Trans. Amer. Math. Soc., 120 (1965), 510-525. | MR | Zbl
,[11] Lp-densities and boundary behaviors of Green potentials, Indiana Univ. Math. J., 28 (1979), 895-911. | Zbl
,[12] Extremal length as a capacity, Michigan Math. J., 17 (1970), 117-128. | MR | Zbl
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