Sharp capacity estimates in s-John domains

It is well-known that several problems related to analysis on $s$-John domains can be unified by certain capacity lower estimates. In this paper, we obtain general lower bounds of $p$-capacity of a compact set $E$ and the central Whitney cube $Q_0$ in terms of the Hausdorff $q$-content of $E$ in an $s$-John domain $\Omega$. Moreover, we construct several examples to show the essential sharpness of our estimates.