Convergence on almost every line for functions with gradient in ${L}^{p}\left({𝐑}^{n}\right)$
Annales de l'Institut Fourier, Tome 24 (1974) no. 3, pp. 159-164.

On démontre que si $\mathrm{grad}\phantom{\rule{0.166667em}{0ex}}\left(f\right)\in {L}^{p}\left({R}^{n}\right)$ pour certaines valeurs de $p$, alors

 $\underset{{x}_{1}\to \infty }{lim}f\left({x}_{1},{x}_{2},...,{x}_{n}\right)=\phantom{\rule{3.33333pt}{0ex}}\text{const.,}\phantom{\rule{4pt}{0ex}}\text{p.p.}\phantom{\rule{4pt}{0ex}}\text{dans}\phantom{\rule{3.33333pt}{0ex}}{R}^{n-1}.$

We prove that if $\mathrm{grad}\phantom{\rule{0.166667em}{0ex}}\left(f\right)\in {L}^{p}\left({R}^{n}\right)$ for certain values of $p$, then

 $\underset{{x}_{1}\to \infty }{lim}f\left({x}_{1},{x}_{2},...,{x}_{n}\right)=\phantom{\rule{3.33333pt}{0ex}}\text{const.,}\phantom{\rule{4pt}{0ex}}\text{a.e.}\phantom{\rule{4pt}{0ex}}\text{in}\phantom{\rule{3.33333pt}{0ex}}{R}^{n-1}.$

@article{AIF_1974__24_3_159_0,
author = {Fefferman, Charles},
title = {Convergence on almost every line for functions with gradient in $L^p({\bf R}^n)$},
journal = {Annales de l'Institut Fourier},
pages = {159--164},
publisher = {Institut Fourier},
volume = {24},
number = {3},
year = {1974},
doi = {10.5802/aif.523},
zbl = {0292.26013},
mrnumber = {52 #11574},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.523/}
}
Fefferman, Charles. Convergence on almost every line for functions with gradient in $L^p({\bf R}^n)$. Annales de l'Institut Fourier, Tome 24 (1974) no. 3, pp. 159-164. doi : 10.5802/aif.523. http://www.numdam.org/articles/10.5802/aif.523/

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