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

Annales de l'institut Fourier (1984)

- Volume: 34, Issue: 1, page 99-109
- ISSN: 0373-0956

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topMizuta, Yoshihiro. "On the boundary limits of harmonic functions with gradient in $L^p$." Annales de l'institut Fourier 34.1 (1984): 99-109. <http://eudml.org/doc/74623>.

@article{Mizuta1984,

abstract = {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 $L^n(\{\bf R\}^n_+)$, $\{\bf R\}^n_+$ denoting the upper half space of the $n$-dimensional euclidean space $\{\bf R\}^n$. 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).},

author = {Mizuta, Yoshihiro},

journal = {Annales de l'institut Fourier},

keywords = {tangential boundary behaviors; harmonic functions with gradient in Lebesgue classes; integral representation},

language = {eng},

number = {1},

pages = {99-109},

publisher = {Association des Annales de l'Institut Fourier},

title = {On the boundary limits of harmonic functions with gradient in $L^p$},

url = {http://eudml.org/doc/74623},

volume = {34},

year = {1984},

}

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

PB - Association des Annales de l'Institut Fourier

VL - 34

IS - 1

SP - 99

EP - 109

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

LA - eng

KW - tangential boundary behaviors; harmonic functions with gradient in Lebesgue classes; integral representation

UR - http://eudml.org/doc/74623

ER -

## References

top- [1] L. CARLESON, Selected Problems on exceptional sets, Van Nostrand, Princeton, 1967. Zbl0189.10903MR37 #1576
- [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. Zbl0495.31003MR83g:31006
- [3] N.G. MEYERS, A theory of capacities for potentials in Lebesgue classes, Math. Scand., 26 (1970), 255-292. Zbl0242.31006MR43 #3474
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- [6] Y. MIZUTA, Existence of various boundary limits of Beppo Levi functions of higher order, Hiroshima Math. J., 9 (1979), 717-745. Zbl0475.31004MR81d:31013
- [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. Zbl0531.31007MR84a:31002
- [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. Zbl0207.13501MR44 #7280
- [10] H. WALLIN, On the existence of boundary values of a class of Beppo Levi functions, Trans. Amer. Math. Soc., 120 (1965), 510-525. Zbl0139.06301MR32 #5911
- [11] J.-M. G. WU, Lp-densities and boundary behaviors of Green potentials, Indiana Univ. Math. J., 28 (1979), 895-911. Zbl0449.31003
- [12] W.P. ZIEMER, Extremal length as a capacity, Michigan Math. J., 17 (1970), 117-128. Zbl0183.39104MR42 #3299

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