Abstract
The paper deals with the theory of inner (outer) capacities on locally compact spaces with respect to general function kernels, the main emphasis being placed on the establishment of alternative characterizations of inner (outer) capacities and inner (outer) capacitary measures for arbitrary sets. The analysis is substantially based on the close interaction between the theory of capacities and that of balayage. As a by-product, we provide a rigorous justification of Fuglede’s theories of inner and outer capacitary measures and capacitability (Acta Math., 1960). The results obtained are largely new even for the logarithmic, Newtonian, Green, α-Riesz, and α-Green kernels.
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The author is deeply indebted to Bent Fuglede for reading and commenting on the manuscript.
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In memory of Makoto Ohtsuka (1922–2007)
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Zorii, N. On the Theory of Capacities on Locally Compact Spaces and its Interaction with the Theory of Balayage. Potential Anal 59, 1345–1379 (2023). https://doi.org/10.1007/s11118-022-10010-3
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DOI: https://doi.org/10.1007/s11118-022-10010-3
Keywords
- Inner and outer capacities and capacitary measures
- Inner and outer balayage
- Capacitability
- Perfect kernels
- Maximum principles
- Principle of positivity of mass