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

Let L α q (R D ),α>0,1<q<, denote the space of Bessel potentials f=G α *g, gL 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<, l’espace des potentiels de Bessel f=G α *g, gL 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.

     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},
     mrnumber = {85f:31015},
     zbl = {0508.31008},
     language = {en},
     url = {}
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Hedberg, Lars-Inge; Wolff, Thomas H. Thin sets in nonlinear potential theory. Annales de l'Institut Fourier, Volume 33 (1983) no. 4, pp. 161-187. doi : 10.5802/aif.944.

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