# Thin sets in nonlinear potential theory

Lars-Inge Hedberg; Thomas H. Wolff

Annales de l'institut Fourier (1983)

- Volume: 33, Issue: 4, page 161-187
- ISSN: 0373-0956

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topHedberg, Lars-Inge, and Wolff, Thomas H.. "Thin sets in nonlinear potential theory." Annales de l'institut Fourier 33.4 (1983): 161-187. <http://eudml.org/doc/74604>.

@article{Hedberg1983,

abstract = {Let $L^q_\alpha (R^D),~\alpha >0,\, 1< q< \infty $, denote the space of Bessel potentials $f=G_\alpha * g$, $g\in L^q$, with norm $\Vert f\Vert _\{\alpha ,q\}=\Vert g\Vert _q$. For $\alpha $ integer $L^q_\alpha $ 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^q_\alpha $, 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 $q>1$.},

author = {Hedberg, Lars-Inge, Wolff, Thomas H.},

journal = {Annales de l'institut Fourier},

keywords = {thin sets; Kellogg and Choquet properties; Wiener criterion; nonlinear potentials},

language = {eng},

number = {4},

pages = {161-187},

publisher = {Association des Annales de l'Institut Fourier},

title = {Thin sets in nonlinear potential theory},

url = {http://eudml.org/doc/74604},

volume = {33},

year = {1983},

}

TY - JOUR

AU - Hedberg, Lars-Inge

AU - Wolff, Thomas H.

TI - Thin sets in nonlinear potential theory

JO - Annales de l'institut Fourier

PY - 1983

PB - Association des Annales de l'Institut Fourier

VL - 33

IS - 4

SP - 161

EP - 187

AB - Let $L^q_\alpha (R^D),~\alpha >0,\, 1< q< \infty $, denote the space of Bessel potentials $f=G_\alpha * g$, $g\in L^q$, with norm $\Vert f\Vert _{\alpha ,q}=\Vert g\Vert _q$. For $\alpha $ integer $L^q_\alpha $ 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^q_\alpha $, 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 $q>1$.

LA - eng

KW - thin sets; Kellogg and Choquet properties; Wiener criterion; nonlinear potentials

UR - http://eudml.org/doc/74604

ER -

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