On the boundary limits of harmonic functions with gradient in $L^p$

Mizuta, Yoshihiro

Annales de l'Institut Fourier, Volume 34 (1984) no. 1, pp. 99-109.

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 $L^n({\mathbf{R}}_{+}^n)$, ${\mathbf{R}}_{+}^n$ denoting the upper half space of the $n$-dimensional euclidean space ${\mathbf{R}}^n$. 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).

Dans cet article on étudie l’allure tangentielle à la frontière des fonctions harmoniques dans la classe de Sobolev $W_1^p({\mathbf{R}}_{+}^n)$, où ${\mathbf{R}}_{+}^n$ est le demi-espace de ${\mathbf{R}}^n$. 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).

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