What is the relationship between the definitions of $s$-capacity (page 13 [here][1]) and $p$-capacity ([here][2]) of a set? 

Are they equivalent? If not, what inequalities hold? What is the difference (in terms of applications) between them?

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For convenience of the reader:

 - **s-capacity**
Define $$I_s(\mu) = \int \int |x-y|^{-s} d\mu(x) d\mu(y).$$
If $A\subset\mathbb{R}^n$ we define the quantity
$$
\mathrm{Cap}_s(A) = \sup \left\{I_s(\mu)^{-1}: \mu \text{ is a finite radon measure on $A$ and } \mu(A) = 1\right\} 
$$
as the $s$-capacity of $A$.

 - **p-capacity** 
Fix $1\leq p<n$. Define,
\begin{equation}
K^p\equiv\{f:\mathbb{R}^n \rightarrow \mathbb{R}\ \vert\ f\geq 0, f\in L^{p^{\ast}}(\mathbb{R}^n), Df\in L^{p}(\mathbb{R}^n;\mathbb{R}^n)\}.
\end{equation}
If $A\subset\mathbb{R}^n$ we define the quantity
\begin{equation}
\text{Cap}_p(A)    \equiv   \inf\left\{\int_{\mathbb{R}^n}\vert Df\vert^p\text{ d}x\ \middle|\   f\in K^p, A\subset\text{int}\{f\geq 1\}\right\}
\end{equation}
as the $p$-capacity of $A$.


  [1]: https://www.math.unipd.it/~monti/PAPERS/Frequency2012.pdf
  [2]: https://math.stackexchange.com/questions/376928/the-definition-of-p-capacity-of-a-set-a-subset-mathbbrn