Decay of the Fourier transform of surfaces with vanishing curvature

L. Erdös, M. Salmhofer, Mathematische Zeitschrift 257 (2007) 261–294.

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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 μ ∧ ε 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)= ∑13 [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.
Publishing Year
Date Published
2007-01-01
Journal Title
Mathematische Zeitschrift
Volume
257
Issue
2
Page
261 - 294
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Erdös L, Salmhofer M. Decay of the Fourier transform of surfaces with vanishing curvature. Mathematische Zeitschrift. 2007;257(2):261-294. doi:10.1007/s00209-007-0125-4
Erdös, L., & Salmhofer, M. (2007). Decay of the Fourier transform of surfaces with vanishing curvature. Mathematische Zeitschrift, 257(2), 261–294. https://doi.org/10.1007/s00209-007-0125-4
Erdös, László, and Manfred Salmhofer. “Decay of the Fourier Transform of Surfaces with Vanishing Curvature.” Mathematische Zeitschrift 257, no. 2 (2007): 261–94. https://doi.org/10.1007/s00209-007-0125-4.
L. Erdös and M. Salmhofer, “Decay of the Fourier transform of surfaces with vanishing curvature,” Mathematische Zeitschrift, vol. 257, no. 2, pp. 261–294, 2007.
Erdös L, Salmhofer M. 2007. Decay of the Fourier transform of surfaces with vanishing curvature. Mathematische Zeitschrift. 257(2), 261–294.
Erdös, László, and Manfred Salmhofer. “Decay of the Fourier Transform of Surfaces with Vanishing Curvature.” Mathematische Zeitschrift, vol. 257, no. 2, Springer, 2007, pp. 261–94, doi:10.1007/s00209-007-0125-4.

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