Soit , l’espace des potentiels de Bessel , , avec la norme . Pour entier peut être identifié à l’espace de Sobolev .
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 , 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 , 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 , démontré antérieurement par l’un des auteurs pour , s’étend au cas .
Let , denote the space of Bessel potentials , , with norm . For integer can be identified with the Sobolev space .
One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space , 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 , 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 , previously proved by one of the authors for , extends to .
@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}, volume = {33}, year = {1983}, pages = {161-187}, doi = {10.5802/aif.944}, mrnumber = {85f:31015}, zbl = {0508.31008}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_1983__33_4_161_0} }
Hedberg, Lars-Inge; Wolff, Thomas H. Thin sets in nonlinear potential theory. Annales de l'Institut Fourier, Tome 33 (1983) pp. 161-187. doi : 10.5802/aif.944. http://gdmltest.u-ga.fr/item/AIF_1983__33_4_161_0/
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