  For small range of $p>2$, we improve the $L^p$ bounds of eigenfunctions of
the Laplacian on negatively curved manifolds. Our improvement is by a power of
logarithm for a full density sequence of eigenfunctions. We also derive
improvements on the size of the nodal sets. Our proof is based on a quantum
ergodicity property of independent interest, which holds for families of
symbols supported in balls whose radius shrinks at a logarithmic rate.
