Decay of the Fourier transform of surfaces with vanishing curvature

We prove L p -bounds on the Fourier transform of measures μ supported on two dimensional surfaces. Our method allows to consider surfaces whose Gauss curvature vanishes on a one-dimensional submanifold. Under a certain non-degeneracy condition, we prove that μ ∧ ε L 4+β, β > 0, and we give a logarithmically divergent bound on the L 4-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, e(p)= ∑13 [1-\cos p_j]} , of the discrete Laplace operator on the cubic lattice. We briefly explain our motivation for this bound originating in the theory of random Schrödinger operators.