L p Norms of Eigenfunctions in the Completely Integrable Case

The eigenfunctions e^iál,x? e^{i\langle\lambda,x\rangle} 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 || j_k || L^p 3 C(e)l_k^{(p-2)/(4p)]-e} \| \varphi_k \| L^{p} \geq C(\epsilon)\lambda_{k}^{{p-2\over4p}-\epsilon} 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.