capacity of a point. I am trying to understand the concept of $(p,\mu)$-capacity, which in the book by Heinonen, Kilpelainen and Martio, the capacity of a compact set $K \subset \Omega$  is defined by:
$$cap_{p,\mu}(K,\Omega) = \inf_{u\in W(K,\Omega)} \int_{\Omega}{|\nabla u|^{p}}d\mu$$
Where, $W(K,\Omega)=\{u \in C_{0}^{\infty}(\Omega): u \geq 1\ on\ K\}$
The authors mention that throughout the whole book, $\Omega$ is an open subset of $R^{n}$ and $n \geq 2$. 
I am trying to understand what is the capacity of a point. I know and I think it  is not hard to prove  that the capacity of a point is zero as long as $1<p \leq n$. However, I read in a paper that if $p>n$ the capacity of a point is always greater than zero.
Can somebody help me to understand why this happens?
I know that for $p>n=1$ any function in the Sobolev space $W^{1,p}((a,b))$ with (a,b) bounded is a.e equal to an abosolutely continuous function. However, I don't know how to use that to show that in the one dimensional case (or more general in $p>n$ case) the capacity of a point is positive.
Thank you
 A: The point is that if $p>n$ then the Sobolev embedding $W^{1,p}_0\hookrightarrow C^{0,\gamma}$, with $\gamma=1-\frac{n}{p}$, is continuous (I am assuming $\mu$ is the Lebesgue measure, otherwise for general measures both the embedding and the result you ask fail). Since trivially also $C^{0,\gamma}\hookrightarrow C^0$ is continuous, we have a continuous embedding $W^{1,p}_0\hookrightarrow C^0$. This means that, given $u\in C^\infty_0(\Omega)$, we have
$$ \|u\|_{C^0(\Omega)}=\|u\|_\infty\leq M \|u\|_{W^{1,p}_0(\Omega)}.
$$
By the  Poincaré inequality the $W^{1,p}$-norm on $W^{1,p}_0$ is equivalent to $\|\nabla u\|_{L^p(\Omega)}$, therefore we have
$$\|u\|_\infty\leq \tilde M \left(\int\limits_\Omega|\nabla u|^p\right)^{1/p}
$$
for any $u\in C^\infty_0(\Omega)$. In particular for any nonempty $K$ and $u\in W(K,\Omega)$ the left hand side is greater than 1, so that the infimum in the definition of capacity gives a positive number.
In dimension one actually the result holds for $p\geq 1$, and there's a simple argument to show it: call $c$ the point in $(a,b)$ which you want to compute the capacity of. Then given $u\in C^\infty_0(a,b)$
$$\int_a^b |u'|=\int_a^c|u'|+\int_c^b|u'|\geq \left|\int_a^c u'\right|+\left|\int_c^b u'\right|=|u(c)-u(a)|+|u(b)-u(c)|\geq 2.
$$
