Trace estimates and improved pointwise bounds for joint eigenfunctions

For $L^2$-normalized joint eigenfunctions in a quantum integrable system, [GT20] gave polynomial improvements over the standard Hömander bounds for typical points. In this paper, we improve their result by establishing a sharp bound of $h^{\frac{-n+k+1}{2}}$ for the points satisfying a rank $k$ non-degeneracy condition.