  We give potential theoretic estimates for the probability that a set $A$
contains a double point of planar Brownian motion run for unit time. Unlike the
probability for $A$ to intersect the range of a Markov process, this cannot be
estimated by a capacity of the set $A$. Instead, we introduce the notion of a
capacity with respect to two gauge functions simultaneously. We also give a
polar decomposition of $A$ into a set that never intersects the set of Brownian
double points and a set for which intersection with the set of Brownian double
points is the same as intersection with the Brownian path.
