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

Mizuta, Yoshihiro

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 $W_1^p({\mathbf{R}}_{+}^n)$, 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).

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).

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