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

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.

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.

@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},
     mrnumber = {85f:31015},
     zbl = {0508.31008},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.944/}
}
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
SP  - 161
EP  - 187
VL  - 33
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.944/
DO  - 10.5802/aif.944
LA  - en
ID  - AIF_1983__33_4_161_0
ER  - 
%0 Journal Article
%A Hedberg, Lars-Inge
%A Wolff, Thomas H.
%T Thin sets in nonlinear potential theory
%J Annales de l'Institut Fourier
%D 1983
%P 161-187
%V 33
%N 4
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.944/
%R 10.5802/aif.944
%G en
%F AIF_1983__33_4_161_0
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. http://archive.numdam.org/articles/10.5802/aif.944/

[1] D. R. Adams and L. I. Hedberg, Inclusion relations among fine topologies in non-linear potential theory, Indiana Univ. Math. J., to appear. | Zbl

[2] D. R. Adams and N. G. Meyers, Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972), 169-197. | MR | Zbl

[3] T. Bagby, Quasi topologies and rational approximation, J. Funct. Anal., 10 (1972), 259-268. | MR | Zbl

[4] A. S. Besicovitch, A general form of the covering principle and relative differentiation of additive functions I, II. Proc. Cambridge Philos. Soc., 41 (1945), 103-110, ibid., 42 (1946), 1-10. | Zbl

[5] M. Brelot, Sur les ensembles effilés, Bull. Sci. Math., 68 (1944), 12-36. | MR | Zbl

[6] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., 175, Springer Verlag 1971. | MR | Zbl

[7] A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., 4 (1961), 33-49. | MR | Zbl

[8] L. Carleson, Selected problems on exceptional sets, Van Nostrand, 1967. | MR | Zbl

[9] G. Choquet, Sur les points d'effilement d'un ensemble. Application à l'étude de la capacité, Ann. Inst. Fourier, Grenoble, 9 (1959), 91-101. | Numdam | MR | Zbl

[10] G. Choquet, Convergence vague et suites de potentiels newtoniens, Bull. Sci. Math., 99 (1975), 157-164. | MR | Zbl

[11] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414. | JFM | MR | Zbl

[12] O. Frostman, Les points irréguliers dans la théorie du potentiel et le critère de Wiener, Kungl. Fysiogr. Sällsk. i Lund Förh., 9-2 (1939), 1-10. | JFM | Zbl

[13] B. Fuglede, Quasi topology and fine topology, Séminaire de Théorie du Potentiel, 10 (1965-1966), no. 12. | Numdam | Zbl

[14] B. Fuglede, The quasi topology associated with a countably additive set function, Ann. Inst. Fourier, Grenoble, 21-1 (1971), 123-169. | Numdam | Zbl

[15] K. Hansson, Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102. | MR | Zbl

[16] V. P. Havin, Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR, 178 (1968), 1025-1028. | MR | Zbl

[17] L. I. Hedberg, Non-linear potentials and approximation in the mean by analytic functions, Math. Z., 129 (1972), 299-319. | MR | Zbl

[18] L. I. Hedberg, Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81. | MR | Zbl

[19] L. I. Hedberg, Spectral synthesis and stability in Sobolev spaces, in Euclidean harmonic analysis (Proc., Univ. of Maryland, 1979), Lecture Notes in Math., 779, 73-103, Springer Verlag 1980. | Zbl

[20] L. I. Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math., 147 (1981), 237-264. | MR | Zbl

[21] L. I. Hedberg, On the Dirichlet problem for higher order equations, in Conference on Harmonic Analysis in Honor of Antoni Zygmund (Chicago 1981), 620-633. Wadsworth, 1983. | Zbl

[22] T. Kolsrud, A uniqueness theorem for higher order elliptic partial differential equations, Math. Scand., 51 (1982), 323-332. | MR | Zbl

[23] N. S. Landkof, Foundations of modern potential theory, Nauka, Moscow 1966. (English translation, Springer-Verlag 1972). | Zbl

[24] V. G. Maz'Ja and V. P. Havin, Non-linear potential theory, Uspehi Mat. Nauk, 27-6 (1972), 67-138. | Zbl

[25] N. G. Meyers, Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR | Zbl

[26] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. | MR | Zbl

Cité par Sources :