Kakeya-Nikodym norms of Maass forms on $\mathrm{U}(2,1)$

Let $\psi$ be a Hecke-Maass form with a large spectral parameter on a compact arithmetic complex hyperbolic surface. We apply the amplification method to obtain a power saving over the trivial bound for the Kakeya-Nikodym norm of $\psi$. As a consequence, we obtain power savings over the local bound of Sogge for $\mathrm{\parallel }\psi {\mathrm{\parallel }}_p$ when $2<p<10\mathrm{/}3$.