# Relationship between $p$-capacity and Riesz $s$-capacity of a set

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?

• 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 $$\mathrm{Cap}_s(A) = \sup \left\{I_s(\mu)^{-1}: \mu \text{ is a finite radon measure and } \mu(\mathbb{R}^n) = 1\right\}$$ as the $$s$$-capacity of $$A$$.
Fix $$1\leq p. Define, $$$$K^p\equiv\{f:\mathbb{R^n}\rightarrow\mathbb{R}\ \vert\ f\geq 0, f\in L^{p^{\ast}}(\mathbb{R}^n), Df\in L^{p}(\mathbb{R}^n;\mathbb{R}^n)\}.$$$$ If $$A\subset\mathbb{R}^n$$ we define the quantity $$$$\text{Cap}_p(A) \equiv \inf\left\{\int_{\mathbb{R}^n}\vert Df\vert^p\text{ d}x\ \middle|\ f\in K^p, A\subset\text{int}\{f\geq 1\}\right\}$$$$ as the $$p$$-capacity of $$A$$.