Relations between the generalized capacity of a condenser in the sense of Aikawa-Ohtsuka and the module of the family of compound curves connecting the condenser plates through a given set are established. Conditions of the removability of a compact set for the generalized capacity of a condenser are obtained. Properties of the extremal length of vector measures are used. Bibliography: 9 titles.
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References
K. Kuratowski, Topology, Vol. II [Russian translation], Mir, Moscow (1969).
P. A. Pugach and V. A. Shlyk, “Generalized capacities and polyhedral surfaces,” Zap. Nauchn. Semin. POMI, 383, 148–179 (2010).
P. A. Pugach and V. A. Shlyk, “Removable sets for the generalized module of a surface family,” Zap. Nauchn. Semin. POMI, 392, 163–190 (2011).
V. A. Shlyk, “Normal domains and removable singularities,” Izv. RAN, Ser. Mat., 57, 93–117 (1993).
H. Aikawa and M. Ohtsuka, “Extremal length of vector measures,” Ann. Acad. Sci. Fenn. Math., 24, 61–88 (1999).
B. Fuglede, “Extremal length and functional completion,” Acta Math., 126, 171–219 (1957).
L. I. Hedberg, “Removable singularities and condenser capacities,” Ark. Mat., 12, 181–201 (1974).
8 B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal functions,” Trans. Amer. Math. Soc., 192, 207–226 (1972).
M. Ohtsuka, Extremal Length and Precise Functions (GAKUTO Int. Ser. Math. Sci. Appl., 19), Gakkōtosho, Tokyo (2003).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 404, 2012, pp. 100–119.
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Dymchenko, Y.V., Shlyk, V.A. Generalized capacities, compound curves, and removable sets. J Math Sci 193, 55–65 (2013). https://doi.org/10.1007/s10958-013-1433-3
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DOI: https://doi.org/10.1007/s10958-013-1433-3