Let , denote the space of Bessel potentials , , with norm . For integer can be identified with the Sobolev space .
One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space , 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 , 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 , previously proved by one of the authors for , extends to .
Soit , l’espace des potentiels de Bessel , , avec la norme . Pour entier peut être identifié à l’espace de Sobolev .
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 , 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 , 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 , démontré antérieurement par l’un des auteurs pour , s’étend au cas .
@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}, zbl = {0508.31008}, mrnumber = {85f:31015}, language = {en}, url = {https://aif.centre-mersenne.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 - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 33 (1983) no. 4, pp. 161-187. doi : 10.5802/aif.944. https://aif.centre-mersenne.org/articles/10.5802/aif.944/
[1] Inclusion relations among fine topologies in non-linear potential theory, Indiana Univ. Math. J., to appear. | Zbl
and ,[2] Thinness and Wiener criteria for non-linear potentials, Indiana Univ. Math. J., 22 (1972), 169-197. | MR | Zbl
and ,[3] Quasi topologies and rational approximation, J. Funct. Anal., 10 (1972), 259-268. | MR | Zbl
,[4] 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] Sur les ensembles effilés, Bull. Sci. Math., 68 (1944), 12-36. | MR | Zbl
,[6] On topologies and boundaries in potential theory, Lecture Notes in Math., 175, Springer Verlag 1971. | MR | Zbl
,[7] Lebesgue spaces of differentiable functions and distributions, Proc. Symp. Pure Math., 4 (1961), 33-49. | MR | Zbl
,[8] Selected problems on exceptional sets, Van Nostrand, 1967. | MR | Zbl
,[9] 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] Convergence vague et suites de potentiels newtoniens, Bull. Sci. Math., 99 (1975), 157-164. | MR | Zbl
,[11] Uniformly convex spaces, Trans. Amer. Math. Soc., 40 (1936), 396-414. | JFM | MR | Zbl
,[12] 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] Quasi topology and fine topology, Séminaire de Théorie du Potentiel, 10 (1965-1966), no. 12. | Numdam | Zbl
,[14] The quasi topology associated with a countably additive set function, Ann. Inst. Fourier, Grenoble, 21-1 (1971), 123-169. | Numdam | Zbl
,[15] Imbedding theorems of Sobolev type in potential theory, Math. Scand., 45 (1979), 77-102. | MR | Zbl
,[16] Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR, 178 (1968), 1025-1028. | MR | Zbl
,[17] Non-linear potentials and approximation in the mean by analytic functions, Math. Z., 129 (1972), 299-319. | MR | Zbl
,[18] Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81. | MR | Zbl
,[19] 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] Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math., 147 (1981), 237-264. | MR | Zbl
,[21] 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] A uniqueness theorem for higher order elliptic partial differential equations, Math. Scand., 51 (1982), 323-332. | MR | Zbl
,[23] Foundations of modern potential theory, Nauka, Moscow 1966. (English translation, Springer-Verlag 1972). | Zbl
,[24] Non-linear potential theory, Uspehi Mat. Nauk, 27-6 (1972), 67-138. | Zbl
and ,[25] Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. | MR | Zbl
,[26] Singular integrals and differentiability properties of functions, Princeton Univ. Press, 1970. | MR | Zbl
,Cited by Sources: