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heat potential; supertemperature; Green function; Riesz measure
We take some well-known inequalities for Green functions relative to Laplace's equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set $E$ whose supports are compact polar subsets of $E$. We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set $E$, we prove a formula for the associated Riesz measure of any point of $E$ in terms of a limit inferior of the quotient of the supertemperature and the Green function for $E$ with a pole at that point.
[1] Doob, J. L.: Classical Potential Theory and Its Probabilistic Counterpart. Grundlehren der Mathematischen Wissenschaften 262 Springer, New York (1984). MR 0731258 | Zbl 0549.31001
[2] Watson, N. A.: Introduction to Heat Potential Theory. Mathematical Surveys and Monographs 182 American Mathematical Society, Providence (2012). DOI 10.1090/surv/182 | MR 2907452 | Zbl 1251.31001
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