For $n\geq2$, $P\subset\mathbb R^n$ is said to be polar if there is some superharmonic function $u$ which is identically $\infty$ on $P$. These sets are generally thought of as small, but I lack intuition for how small exactly- can anything be said about their Hausdorff dimension, for example? Must it always be less than $n$? At most $n-1$? The texts I have been reading seem a little scarce on examples of polar sets, beyond countable sets and a line in the case $n>1$, and I have yet to properly digest the theory of capacity. I am aware that polar sets have Lebesgue measure zero, so asking whether their Hausdorff dimension is less than $n$ seems reasonable.
Add a comment |
1 Answer
$\begingroup$ $\endgroup$
A polar set in $\mathbb{R}^n$ has Hausdorff dimension not more than $n-2$. This is Theorem 5.9.6 in
D.H. Armitage, S.J. Gardiner, Classical potential theory. Springer Monographs in Mathematics. Springer-Verlag, 2001.
For the special case $n=2$, this is also Theorem III.19 p.65 in
M. Tsuji, Potential theory in modern function theory. Maruzen Co. Ltd., Tokyo, 1959.
