What is the relationship between the definitions of $s$-capacity (page 13 here) and $p$-capacity (here) of a set?
Are they equivalent? If not, what inequalities hold? What is the difference (in terms of applications) between them?
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
$$
\mathrm{Cap}_s(A) = \sup \left\{I_s(\mu)^{-1}: \mu \text{ is a finite radon measure and } \mu(\mathbb{R}^n) = 1\right\}
$$
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 $A$.
as the $s$-capacity of $A$. - p-capacity
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 $A$.
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 $A$.
