Decay of the Fourier transform of surfaces with vanishing curvature

Abstract

We prove $L^p$-bounds on the Fourier transform of measures $\mu$ 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 $\whμ\in L^{4+β}$, $\beta >0$, 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, $e(p)={\sum }_1^3[1-\cos p_j]$, 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.

arXiv comment40 pages, 4 figures

References in corpus (1)

• A Fourier restriction theorem for hypersurfaces which are graphs of certain real polynomials

Cited by in corpus (1)

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