Thin sets in nonlinear potential theory

Hedberg, Lars-Inge; Wolff, Thomas H.

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

Soit $L_{\alpha }^q(R^D),\alpha >0,1<q<\infty$, 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 $L_{\alpha }^q(R^D),\alpha >0,1<q<\infty$, 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 $q>1$.

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