On Fourier restriction and the Newton polygon

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.