Decay of the Fourier transform of surfaces with vanishing curvature

We prove Lp-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 $${\hat{\mu}\in L^{4+\beta}}$$ , β > 0, and we give a logarithmically divergent bound on the L4-norm. We use this latter bound to estimate almost singular integrals involving the dispersion relation, $${e(p)= \sum_1^3 [1-\cos p_j]}$$ , 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.