On weighted extremal length and p-capacity
The paper gives a relation between weighted extremal length and weighted p-capacity for weights w satisfying the Muckenhoupt \(A_ p\) condition. Let \(\Gamma\) be a family of locally rectifiable curves in \({\mathbb{R}}^ d\) and let \(1<p<\infty\). The weighted module of order p of \(\Gamma\) is defined by \[ M_ p(\Gamma;w)=\inf \{\int_{{\mathbb{R}}^ d}\rho^ p w dx;\quad \rho \text{ is \(\Gamma\)-admissible}\} \] and the weighted extremal length by the reciprocal of the weighted module. For a compact set E in \({\mathbb{R}}^ d\) and a domain G containing E the weighted p- capacity of (E,G) is denoted by C(E;G). The following three theorems are proven: Theorem 1. Suppose \({\mathbb{R}}^ d-E\) is a domain. Let G be a bounded domain containing E and let \(\Gamma\) be the family of curves connecting E and \(\partial G\) in G-E. Then \(M_ p(\Gamma;w)=C^ w_ p(E;G).\) Theorem 2. Let \(\Gamma_{\infty}\) be the family of curves in \({\mathbb{R}}^ d-E\) connecting E and the point at infinity. Then \(M_ p(\Gamma_{\infty};w)=C^ w_ p(E).\) Theorem 3. Let \(w\in A_{p,1}\) and let \(\Lambda\) (E) be the family of curves in \({\mathbb{R}}^ d-E\) terminating at points of E. Then \(M_ p(\Lambda (E);w)=0\) if and only if \(C^ w_ p(E)=0\).
scientific article; zbMATH DE number 4198430