zbMATH — the first resource for mathematics

Fine and quasi connectedness in nonlinear potential theory. (English) Zbl 0545.31012
The main results of this paper concern the connectedness of subsets of Euclidean \(n\)-space in the \((\alpha,p)\)-fine and \((\alpha,p)\)-quasi topologies, i.e. the fine and quasi topologies associated with the Bessel capacities \(B_{\alpha,p}\) [see L. I. Hedberg and T. Wolff, Ann. Inst. Fourier 33, No.4, 161-187 (1983; Zbl 0508.31008), and the references cited there]. In particular, the inequality \[ (*)\quad \min \{B_{\alpha,p}(E\cap Q),B_{\alpha,p}(E^ c\cap Q)\}\leq C\cdot B_{\alpha,p}(Q\cap \partial_ fE) \] is established for all cubes \(Q\); here \(E\) is an arbitrary set, \(E^ c\) its complement, and \(\partial_ fE\) the boundary of \(E\) in the \((\alpha,p)\)-fine topology. (\(C\) is a constant depending only on \(\alpha\), \(p\) and \(N\).) From (*) it follows that if \(G\) is an open and connected subset of \({\mathbb{R}}^ N\), then \(G\) is connected in the \((\alpha,p)\)-quasi topology provided \(\alpha p>1\). It also follows from (*) that if \(E\) is both \((\alpha,p)\)-quasi open and \((\alpha,p)\)-quasi closed, then either \(B_{\alpha,p}(E)=0\) or \(B_{\alpha,p}(E^ c)=0\), provided \(\alpha p>1\). All of these corollaries of (*) extend results of B. Fuglede [Ann. Inst. Fourier 21, No.3, 227-244 (1971; Zbl 0208.138), and ibid. 21, No.1, 123-169 (1971; Zbl 0197.194)] who treats the classical case \(\alpha =1\), \(p=2\) using the method of balayage, i.e. the maximum principle. For other values of \(\alpha\) and \(p\) no such method is available, hence the present paper exploits geometric measure theoretic arguments together with the recently established Kellogg and Choquet properties for the capacities \(B_{\alpha,p}\) (for all values of \(p>1)\) [see L. I. Hedberg and T. Wolff, loc. cit.]. The question of arcwise connectedness is also considered and the following theorem is established; it generalizes a result of T. J. Lyons [J. Funct. Anal. 37, 1-18 (1980; Zbl 0459.46038)]. Theorem. If \(E\) is \((\alpha,p)\)-finely open and \((\alpha,p)\)-finely connected, then for \(\alpha p>1\), \(E\) is arcwise connected. This result is false for \(\alpha p\leq 1\).

31D05 Axiomatic potential theory
Full Text: DOI Numdam EuDML
[1] D.R. ADAMS, Traces of potentials. II., Ind. U. Math. J., 22 (1973), 907-918. · Zbl 0265.46039
[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. · Zbl 0545.31011
[4] D.R. ADAMS, and N.G. MEYERS, Thinness and Wiener criteria for non-linear potentials, Ind. U. Math. J., 22 (1972), 169-197. · Zbl 0244.31012
[5] M. BRELOT, On topologies and boundaries in potential theory, Lecture Notes in Math. 175, Springer-Verlag. · Zbl 0222.31014
[6] B. DAVIS and J.L. LEWIS, Paths for subharmonic functions, Proc. London Math. Soc., 48 (1984), 401-427. · Zbl 0541.31001
[7] B. FUGLEDE, Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, 21-3 (1971), 227-244. · Zbl 0208.13802
[8] B. FUGLEDE, The quasi topology associated with a countably subadditive set function, Ann. Inst. Fourier, 21-1 (1971), 123-169. · Zbl 0197.19401
[9] C. FERNSTROM, On the instability of capacity, Ark. Mat., 15 (1971), 241-252. · Zbl 0372.31003
[10] H. FEDERER, Geometric measure theory, Springer-Verlag, 1969. · Zbl 0176.00801
[11] C. GOFFMAN and D. WATERMAN, Approximately continuous transformations, Proc. Amer. Math. Soc., 12 (1961), 116-121. · Zbl 0096.17103
[12] L.I. HEDBERG, Non-linear potentials and approximation in the mean by analytic functions, Math. Z., 129 (1972), 299-319. · Zbl 0236.31010
[13] L.I. HEDBERG and T. WOLFF, Thin sets in non-linear potential theory, Ann. Inst. Fourier, 33-4 (1983), 161-187. · Zbl 0508.31008
[14] T. LYONS, Finely holomorphic functions, J. Func. Anal., 37 (1980), 1-18. · Zbl 0459.46038
[15] V. MAZ’YA and V. HAVIN, Non-linear potential theory, Russian Math. Surveys, 27 (1972), 71-148. · Zbl 0269.31004
[16] N.G. MEYERS, A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand., 26 (1970), 255-292. · Zbl 0242.31006
[17] N.G. MEYERS, Continuity properties of potentials, Duke Math. J., 42 (1975), 157-166. · Zbl 0334.31004
[18] N.G. MEYERS, Continuity of Bessel potentials, Israel J. Math., 11 (1972), 271-283. · Zbl 0256.31009
[19] J. RIDDER, Uber approximativ statige funktionen von zwei (und mehreren) veranderlichen, Fund. Math., 13 (1927), 201-209. · JFM 55.0145.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.