L^p norms of eigenfunctions in the completely integrable case

The eigenfunctions e^i〈λ,x〉 of the Laplacian on a flat torus have uniformly bounded L^p norms. In this article, we prove that for every other quantum integrable Laplacian, the L^p norms of the joint eigenfunctions blow up at least at the rate ||φ_k||_Lp ≥ C(∈)λ_k^p-2/4p-∈when p > 2. This gives a quantitative refinement of our recent result [TZ1] that some sequence of eigenfunctions must blow up in L^p unless (M, g) is flat. The better result in this paper is based on mass estimates of eigenfunctions near singular leaves of the Liouville foliation.