The authors establish some properties of weight spaces \(L^1_{p,\omega}\) and \(FD^{p,\omega}\) [cf. the second author, ibid. 332, 428-431 (1993; Zbl 0863.31007)], where \(\omega \) satisfies the \(A_p\)-condition of B. Muckenhoupt [Trans. Am. Math. Soc. 165, 207-226 (1972; Zbl 0236.26016)]. In (loc. cit.), V. A. Shlyk introduced the concept of \(p\)-capacity \(C_p(F_0, F_1, G/F, \omega )\) (with weight \(\omega \)) of a condenser \((F_0, F_1, G/F)\), where \(F_0, F_1\subset\overline G\) are nonempty, disjoint compacta and \(F\) is a bounded set, closed with respect to \(R^n\setminus (F_0\cup F_1)\), which does not have components with limit points on \(F_0\cup F_1\). Let \(\{\tau _j\} \) be a sequence of curves \(\tau _j\subset\overline{R^n}\) with \(\tau _j\setminus\{\infty\} \) locally rectifiable; if \(\tau _j\subset G/(F_0\cup F_1\cup F)\), \(\overline\tau _j\setminus\tau _j\subset F_0\cup F_1\cup F\) for each \(j, \gamma =\displaystyle\bigcup _j\tau _j\) and \(\gamma\cup F_0\cup F_1\cup F\) contains a continuum joining \(F_0\) with \(F_1\), then \(\gamma \) is called a composite curve joining \(F_0\) and \(F_1\) in \(G/F\). The family of all these curves is denoted by \(\Gamma (F_0, F_1, G/F)\). In (loc. cit.) V. A. Shlyk defined the notion of \(p\)-module with weight \(\omega \) of the family \(\Gamma (F_0, F_1, G/F)\) and established the equality between this \(p\)-module and the \(p\)-capacity from above. These concepts of \(p\)-capacity and \(p\)-module generalize the corresponding ones of the \(p\)-capacity \(\text{cap}_p(F_0, F_1, G)\) of a condenser \((F_0, F_1, G)\) and of the \(p\)-module \(M_p\Gamma (F_0, F_1, G)\) of an arc family \(\Gamma (F_0, F_1, G)\) joining \(F_0\) and \(F_1\) in \(G\).
By means of this result, the authors show that the linear span of the class \(E_{p,\omega}(G)\) of all extremal functions for \(C_p(F_0, F_1, G/\emptyset ,\omega )\) with smooth compacta \(F_0, F_1\subset G\) is everywhere dense in \(L^1_{p,\omega}(G)\). Then, for two open sets \(G_1, G_2\), \((G_1\subset G_2)\), they give some necessary and sufficient conditions for \(G_2\setminus G_1\) to be an \(NC_{p,\omega}\)-set in \(G_2\). (A set \(E\), closed with respect to \(G\), is called an \(NC_{p,\omega}\)-set if for any pair of smooth disjoint compacta \(F_0, F_1\subset G\setminus E\), \(C_p[F_0, F_1, (G\setminus E)/\emptyset ,\omega]=C_p(F_0, F_1, G/\emptyset , \omega )\).) Finally, they obtain necessary and sufficient conditions for a compact \(E\subset G\) to be removable for \(FD^{p,\omega}\). Some of these theorems are given, for the sake of simplicity, in the particular case in which \(E\) is of measure zero.
Reviewer’s remark: I want to point out that in the English version of this note, there are several inappropriate translations of some basic terms and expressions as follows: “capacitor” instead of “condenser”, “able sets” instead of “removable sets” (as in the title), “weight function class \(L^1_{p,\omega}\) and \(FD^{p,\omega}\)” instead of “weight class \(L^1_{p,\omega}\) and \(F^{p,\omega}\) of functions”, “the population of all extremal functions” instead of “the class of all extremal functions”, “is dense everywhere in \(L^1_{p,\omega}\)” instead of “is everywhere dense in \(L^1_{p,\omega}\)” and “zero-dimensional” instead of “of measure zero”.