Thin sets in nonlinear potential theory

Hedberg, Lars-Inge; Wolff, Thomas H.

doi:10.5802/aif.944

Hedberg, Lars-Inge ; Wolff, Thomas H.

Annales de l'Institut Fourier, Tome 33 (1983) no. 4, pp. 161-187

Abstract (VO)
Résumé

Let $L_{α}^{q} (R^{D}), α > 0, 1 < q < \infty$ , denote the space of Bessel potentials $f = G_{α} * g$ , $g \in L^{q}$ , with norm $∥ f ∥_{α, q} = ∥ g ∥_{q}$ . For $α$ integer $L_{α}^{q}$ can be identified with the Sobolev space $H^{α, q}$ .

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 $L_{α}^{q}$ , 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 $H^{α - q}$ , previously proved by one of the authors for $q > 2 - α / d$ , extends to $q > 1$ .

Soit $L_{α}^{q} (R^{D}), α > 0, 1 < q < \infty$ , l’espace des potentiels de Bessel $f = G_{α} * g$ , $g \in L^{q}$ , avec la norme $∥ f ∥_{α, q} = ∥ g ∥_{q}$ . Pour $α$ entier $L_{α}^{q}$ peut être identifié à l’espace de Sobolev $H^{α, q}$ .

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 $L_{α}^{q}$ , 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 $H^{α, q}$ , démontré antérieurement par l’un des auteurs pour $p > 2 - α / d$ , s’étend au cas $q > 1$ .

MR Zbl

DOI : 10.5802/aif.944

@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},
     address = {Grenoble},
     volume = {33},
     number = {4},
     year = {1983},
     doi = {10.5802/aif.944},
     zbl = {0508.31008},
     mrnumber = {85f:31015},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.944/}
}

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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

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