On Fourier restriction and the Newton polygon

Magyar, Ákos

doi:10.1090/S0002-9939-08-09510-5

On Fourier restriction and the Newton polygon
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Proc. Amer. Math. Soc. 137 (2009), 615-625 Request permission

Abstract:

Local $L^p\to L^2$ bounds are proved for the restriction of the Fourier transform to analytic surfaces of the form $S=(x,f(x))$ in $\mathbb {R}^3$. It is found that the range of exponents is determined by the so-called distance of the Newton polygon, associated to $f$, except when the principal quasi-homogeneous part of $f(x)$ contains a factor of high multiplicity. The proofs are based on the method of Phong-Stein and Rychkov, adapted to scalar oscillatory integrals.

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