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Quasi-bounded and singular thermal potentials

Published online by Cambridge University Press:  24 October 2008

N. A. Watson
Affiliation:
University of Canterbury, New Zealand

Extract

In his monograph [2], Doob presented the theories of quasi-bounded and singular functions associated with the Laplace and heat equations. In particular, on p. 712, he proved in the classical case that a potential is quasi-bounded if and only if its associated measure vanishes on polar sets. However, on p. 714, he proved only weaker results for the thermic case, namely that a potential is quasi-bounded (i) if its associated measure vanishes on semi-polar sets, and (ii) only if its measure vanishes on polar sets. The main purpose of this note is to establish that the exact analogue of the result for the classical case holds for the thermic case. We also use a related theorem on thermal potentials, to prove a second characterization of those which are quasi-bounded, analogous to one due to Arsove and Leutwiler for the classical case. We also give the corresponding characterizations of singular thermal potentials.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Arsove, M. and Leutwiler, H.. Quasi-bounded and singular functions. Trans. Amer. Math. Soc. 189 (1974), 275302.CrossRefGoogle Scholar
[2]Doob, J. L.. Classical potential theory and its probabilistic counterpart (Springer-Verlag, 1984).CrossRefGoogle Scholar