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