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Decay of the Fourier transform of surfaces with vanishing curvature

Laszlo Erdos·Manfred Salmhofer
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Abstract

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 $\whμ\in L^{4+β}$, $β>0$, 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, $e(p)= \sum_1^3 [1-\cos p_j]$, 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.

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Laszlo Erdos, Manfred Salmhofer. 2006-04-18. Decay of the Fourier transform of surfaces with vanishing curvature. https://arxiv.org/abs/math-ph/0604039

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