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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 \(\alpha p\leq 1\).

MSC:
31D05 Axiomatic potential theory
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