Thin sets in nonlinear potential theory
Annales de l'Institut Fourier, Tome 33 (1983) no. 4, pp. 161-187.

Soit ${L}_{\alpha }^{q}\left({R}^{D}\right),\phantom{\rule{3.33333pt}{0ex}}\alpha >0,\phantom{\rule{0.166667em}{0ex}}1, l’espace des potentiels de Bessel $f={G}_{\alpha }*g$, $g\in {L}^{q}$, avec la norme $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_{q}$. Pour $\alpha$ entier ${L}_{\alpha }^{q}$ peut être identifié à l’espace de Sobolev ${H}^{\alpha ,q}$.

On peut associer une théorie du potentiel à ces espaces d’une manière semblable à la manière dont la théorie classique du potentiel est associée à l’espace ${H}^{1,2}$, et en large partie la théorie a été étendue à cette situation plus générale autour de 1970. Néanmoins il y avait des problèmes à étendre la théorie des ensembles effilés. Moyennant une nouvelle inégalité, qui caractérise le cône positif dans l’espace dual de ${L}_{\alpha }^{q}$, nous comblons ce manque. Nous montrons qu’il y a une “bonne” définitions des ensembles effilés, telle que les propriétés de Kellogg et de Choquet aient lieu et telle qu’il y ait un critère de Wiener pour certains potentiels non-linéaires.

Comme conséquence de la propriété de Kellogg, le “théorème de synthèse spectrale” pour ${H}^{\alpha ,q}$, démontré antérieurement par l’un des auteurs pour $p>2-\alpha /d$, s’étend au cas $q>1$.

Let ${L}_{\alpha }^{q}\left({R}^{D}\right),\phantom{\rule{3.33333pt}{0ex}}\alpha >0,\phantom{\rule{0.166667em}{0ex}}1, denote the space of Bessel potentials $f={G}_{\alpha }*g$, $g\in {L}^{q}$, with norm $\parallel f{\parallel }_{\alpha ,q}=\parallel g{\parallel }_{q}$. For $\alpha$ integer ${L}_{\alpha }^{q}$ can be identified with the Sobolev space ${H}^{\alpha ,q}$.

One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space ${H}^{1;2}$, and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the positive cone in the space dual to ${L}_{\alpha }^{q}$, we fill this gap. We show that there is a “good” definition of thin sets, such that the Kellogg and Choquet properties hold, and such that there is a Wiener criterion for certain nonlinear potentials.

As a consequence of the Kellogg property the “spectral synthesis theorem” for ${H}^{\alpha -q}$, previously proved by one of the authors for $q>2-\alpha /d$, extends to $q>1$.

@article{AIF_1983__33_4_161_0,
author = {Hedberg, Lars-Inge and Wolff, Thomas H.},
title = {Thin sets in nonlinear potential theory},
journal = {Annales de l'Institut Fourier},
pages = {161--187},
publisher = {Institut Fourier},
volume = {33},
number = {4},
year = {1983},
doi = {10.5802/aif.944},
zbl = {0508.31008},
mrnumber = {85f:31015},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.944/}
}
Hedberg, Lars-Inge; Wolff, Thomas H. Thin sets in nonlinear potential theory. Annales de l'Institut Fourier, Tome 33 (1983) no. 4, pp. 161-187. doi : 10.5802/aif.944. http://www.numdam.org/articles/10.5802/aif.944/

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