Dichotomy of global capacity density

Abstract:

Let $1<p<\infty$ and let $d\mu (x)=w(x)dx$ be a $p$-admissible weight in $\mathbb {R}^n$, $n\ge 2$. By $\mathrm {Cap}_{p,\mu }(E,D)$ we denote the variational $(p,\mu )$-capacity of condenser $(E,D)$. We show a dichotomy of the global density with respect to $\mathrm {Cap}_{p,\mu }$. One of our results is as follows: Let $\lambda >1$ and let $B(x,r)$ stand for the open ball with center at $x$ and radius $r$. Then \[ \lim _{r\to \infty }\Bigg (\inf _{x\in \mathbb {R}^n} \frac {\mathrm {Cap}_{p,\mu }(E\cap B(x,r),B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r),B(x,\lambda r))} \Bigg )\] is equal to either 0 or 1; the first case occurs if and only if \[ \inf _{x\in \mathbb {R}^n} \frac {\mathrm {Cap}_{p,\mu }(E\cap B(x,r_0), B(x,\lambda r))}{\mathrm {Cap}_{p,\mu }(B(x,r), B(x,\lambda r))} \] is identically equal to 0. This provides a sharp contrast between capacity and Lebesgue measure.