Fine and quasi connectedness in nonlinear potential theory
Annales de l'Institut Fourier, Volume 35 (1985) no. 1, pp. 57-73.

If B α,p denotes the Bessel capacity of subsets of Euclidean n-space, α>0, 1<p<, naturally associated with the space of Bessel potentials of L p -functions, then our principal result is the estimate: for 1<αpn, there is a constant C=C(α,p,n) such that for any set E

min { B α , p ( E Q ) , B α , p ( E c Q ) } C · B α , p ( Q f E )

for all open cubes Q in n-space. Here f E is the boundary of the E in the (α,p)-fine topology i.e. the smallest topology on c-space that makes the associated (α,p)-linear potentials continuous there. As a consequence, we deduce that for αp>1, open connected sets are connected in the (α,p)-quasi topology (i.e. the topology generated by the set function B α,p in the sense of Fuglede), and the (α,p)-finely open (α,p)-finely connected sets are arcwise connected. Our methods rely on the Kellog-Choquet properties of the capacities B α,p and aspects of geometric measure theory. The classical Newtonian case corresponds to the case α=1, p=2 and n=3.

Si B α,p désigne la capacité de Bessel des sous-ensembles de l’espace euclicien de dimension n, α>0, 1<p<, associé naturellement avec l’espace des potentiels de Bessel des fonctions L p -functions, alors notre résultat principal est l’estimation suivante : pour 1<αpn, il existe une constante C=C(α,p,n) de telle sorte que pour n’importe quel ensemble E,

min { B α , p ( E Q ) , B α , p ( E c Q ) } C · B α , p ( Q f E )

pour tous les cubes ouverts Q dans l’espace de dimension n. Ici, f E est le bord de l’ensemble E dans la topologie —fine (α,p)— c’est-à-dire la topologie minimale sur l’espace de dimension n qui rend continu les potentiels (α,p)-non-linéaires associés. Par conséquent, nous déduisons que pour αp>1, les ensembles ouverts et connexes sont connexes dans la (α,p)-quasi-topologie (c’est-à-dire la topologie engendrée par la fonction de l’ensemble B α,p au sens de Fuglede) et que les ensembles (α,p)-finement ouverts (α,p)-finement connexes sont connexes par arcs. Nos méthodes sont basées sur les propriétés de Kellog-Choquet des capacités B α,p et certains aspects de la théorie de la mesure géométrique. Le cas newtonien classique correspond au cas α=1, p=2 et n=3.

@article{AIF_1985__35_1_57_0,
     author = {Adams, David R. and Lewis, John L.},
     title = {Fine and quasi connectedness in nonlinear potential theory},
     journal = {Annales de l'Institut Fourier},
     pages = {57--73},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {35},
     number = {1},
     year = {1985},
     doi = {10.5802/aif.998},
     mrnumber = {86h:31009},
     zbl = {0545.31012},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.998/}
}
TY  - JOUR
AU  - Adams, David R.
AU  - Lewis, John L.
TI  - Fine and quasi connectedness in nonlinear potential theory
JO  - Annales de l'Institut Fourier
PY  - 1985
SP  - 57
EP  - 73
VL  - 35
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.998/
DO  - 10.5802/aif.998
LA  - en
ID  - AIF_1985__35_1_57_0
ER  - 
%0 Journal Article
%A Adams, David R.
%A Lewis, John L.
%T Fine and quasi connectedness in nonlinear potential theory
%J Annales de l'Institut Fourier
%D 1985
%P 57-73
%V 35
%N 1
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.998/
%R 10.5802/aif.998
%G en
%F AIF_1985__35_1_57_0
Adams, David R.; Lewis, John L. Fine and quasi connectedness in nonlinear potential theory. Annales de l'Institut Fourier, Volume 35 (1985) no. 1, pp. 57-73. doi : 10.5802/aif.998. http://archive.numdam.org/articles/10.5802/aif.998/

[1] D.R. Adams, Traces of potentials. II., Ind. U. Math. J., 22 (1973), 907-918. | MR | Zbl

[2] D.R. Adams, Lectures on Lp-potential theory, Umeå Univ. Reports, no. 2 (1981).

[3] D.R. Adams and L.I. Hedberg, Inclusion relations among fine topologies in non-linear potential theory, Ind. U. Math. J., 33 (1984), 117-126. | MR | Zbl

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

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

[6] B. Davis and J.L. Lewis, Paths for subharmonic functions, Proc. London Math. Soc., 48 (1984), 401-427. | MR | Zbl

[7] B. Fuglede, Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, 21-3 (1971), 227-244. | EuDML | Numdam | MR | Zbl

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

[9] C. Fernstrom, On the instability of capacity, Ark. Mat., 15 (1971), 241-252. | MR | Zbl

[10] H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. | MR | Zbl

[11] C. Goffman and D. Waterman, Approximately continuous transformations, Proc. Amer. Math. Soc., 12 (1961), 116-121. | MR | Zbl

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

[13] L.I. Hedberg and T. Wolff, Thin sets in non-linear potential theory, Ann. Inst. Fourier, 33-4 (1983), 161-187. | Numdam | MR | Zbl

[14] T. Lyons, Finely holomorphic functions, J. Func. Anal., 37 (1980), 1-18. | MR | Zbl

[15] V. Maz'Ya and V. Havin, Non-linear potential theory, Russian Math. Surveys, 27 (1972), 71-148. | Zbl

[16] N.G. Meyers, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand., 26 (1970), 255-292. | MR | Zbl

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

[18] N.G. Meyers, Continuity of Bessel potentials, Israel J. Math., 11 (1972), 271-283. | MR | Zbl

[19] J. Ridder, Uber approximativ statige Funktionen von zwei (und mehreren) Veranderlichen, Fund. Math., 13 (1927), 201-209. | JFM

Cited by Sources: