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Decay of the Fourier transform of surfaces with vanishing curvature

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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 \({\hat{\mu}\in L^{4+\beta}}\) , 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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Correspondence to László Erdős.

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L. Erdős was partially supported by EU-IHP Network “Analysis and Quantum” HPRN-CT-2002-0027. M. Salmhofer was partially supported by DFG grant Sa 1362/1–1 and an ESI senior research fellowship.

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Erdős, L., Salmhofer, M. Decay of the Fourier transform of surfaces with vanishing curvature. Math. Z. 257, 261–294 (2007). https://doi.org/10.1007/s00209-007-0125-4

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  • DOI: https://doi.org/10.1007/s00209-007-0125-4

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