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In recent research of my coauthors and me, it has become necessary to consider the Riesz potential of a superadditive measure.

Recall that the $s$-dimensional Riesz potential of a finite Borel measure $\mu$ on $\mathbb{R}^d$ is

$$U^s_{\mu}(x) = \int |x-y|^{-s}\,d\mu(y).$$

We extended this potential operator to certain classes of superadditive measures. (Here, superadditive means $\mu(A)+\mu(B) \leq \mu (A \cup B)$ for disjoint $A$ and $B$.)

Is this something that has been investigated before?

Are there any known applications?

(Either way, we were able to solve the problem we set out to solve, but it would be nice to know if this is connected to something that has been previously looked at.)

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