Decay of the Fourier transform of surfaces with vanishing curvature

Details Decay of the Fourier transform of surfaces with vanishing curvature Decay of the Fourier transform of surfaces with vanishing curvature

2006-04-18

arxiv.org/abs/math-ph/0604039v1

Physics

Mathematical Physics

40 pages, 4 figures

Scientific paper

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 $\wh\mu\in L^{4+\beta}$, $\beta>0$, 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 Schr\"odinger operators.

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