Fine and quasi connectedness in nonlinear potential theory

Adams, David R.; Lewis, John L.

Annales de l'Institut Fourier, Tome 35 (1985) no. 1, pp. 57-73.

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

$$min\{B_{\alpha ,p}(E\cap Q),B_{\alpha ,p}(E^c\cap Q\left)\right\}\le C·B_{\alpha ,p}(Q\cap {\partial }_{f}E)$$

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

If $B_{\alpha ,p}$ denotes the Bessel capacity of subsets of Euclidean $n$-space, $\alpha >0$, $1<p<\infty$, naturally associated with the space of Bessel potentials of $L^p$-functions, then our principal result is the estimate: for $1<\alpha p\le n$, there is a constant $C=C(\alpha ,p,n)$ such that for any set $E$

$$min\{B_{\alpha ,p}(E\cap Q),B_{\alpha ,p}(E^c\cap Q\left)\right\}\le C·B_{\alpha ,p}(Q\cap {\partial }_{f}E)$$

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

[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