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

Abstract

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Let L α q ( R D ) , α > 0 , 1 < q < ∞ , denote the space of Bessel potentials f = G α * g , g ∈ 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 .

How to cite

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Hedberg, 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 &gt;0,\, 1&lt; q&lt; \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 $q&gt;2-\alpha /d$, extends to $q&gt;1$.},
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 &gt;0,\, 1&lt; q&lt; \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 $q&gt;2-\alpha /d$, extends to $q&gt;1$.
LA - eng
KW - thin sets; Kellogg and Choquet properties; Wiener criterion; nonlinear potentials
UR - http://eudml.org/doc/74604
ER -

References

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Citations in EuDML Documents

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  1. J. H. Michael, William P. Ziemer, The Wiener criterion and quasilinear uniformly elliptic equations
  2. Jan Malý, The area formula for W 1 , n -mappings
  3. L. Boccardo, D. Giachetti, F. Murat, A generalization of a theorem of H. Brezis & F. E. Browder and applications to some unilateral problems
  4. David R. Adams, John L. Lewis, Fine and quasi connectedness in nonlinear potential theory
  5. Tero Kilpeläinen, Jan Malý, Degenerate elliptic equations with measure data and nonlinear potentials
  6. Jan Malý, Pointwise estimates of nonnegative subsolutions of quasilinear elliptic equations at irregular boundary points
  7. Juha Heinonen, Terro Kilpeläinen, Olli Martio, Fine topology and quasilinear elliptic equations
  8. Enrico Vitali, Convergence of unilateral convex sets in higher order Sobolev spaces
  9. Verbitsky, Igor E., Superlinear equations, potential theory and weighted norm inequalities
  10. Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Improved estimates for the Ginzburg-Landau equation : the elliptic case

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