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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
##### Citations:
Zbl 0508.31008; Zbl 0208.138; Zbl 0197.194; Zbl 0459.46038
Full Text:
##### References:
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