  For $f: [0,1]\to \R$, we consider $L^f_t$, the local time of space-time
Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions
whose H\"older norm of order $\al$ is less than or equal to 1. We show that the
supremum of $L^f_1$ over $f$ in $\sS_\al$ is finite is $\al>\frac12$ and
infinite if $\al<\frac12$.
