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The Fourier restriction problem for a submanifold $S$ in $\mathbb{R}^n$ asks for which exponents $p, q \in [1,\infty]$ the Fourier restriction operator $R$ is bounded from $L^p(\mathbb{R}^n, \mathrm{d}x)$ to $L^q(S, \rho \mathrm{d}S)$. Here $\mathrm{d}S$ denotes the Riemannian surface measure of $S$ and $\rho$ a fixed $C_c^\infty(S)$ function. Aside from being one of the central problems in harmonic analysis the Fourier restriction problem plays an important role in a variety of other areas of mathematics such as the theory of nonlinear dispersive equations, geometric measure theory, and number theory. It was originally introduced by E. M. Stein around 1970 and since then a lot of deep work has been done on this problem by many renowned mathematicians, including Fields Medalists C. Fefferman, J. Bourgain, and T. Tao. Though the restriction problem for curves with nonvanishing curvature in $\mathbb{R}^2$ was already solved in the early 1970s through contributions by C. Fefferman, E. M. Stein, and A. Zygmund, it remains wide open even for the sphere in $\mathbb{R}^3$. The first result for higher dimensions, obtained by P. A. Tomas and E. M. Stein in the 1970s, was a sharp $L^p - L^2$ restriction estimate for the unit sphere in $\mathbb{R}^n$. These $L^p - L^2$ restriction estimates (also called Stein-Tomas type estimates) are much easier to handle than the general $L^p - L^q$ estimates. Indeed, by the $R^* R$ method they reduce to $L^{p} - L^{p'}$ estimates for the convolution operator with integral kernel $\widehat{\rho \mathrm{d}S}$. If we can locally represent $S$ as a graph of a function $\phi$, then the integral kernel can be written as an oscillatory integral having $phi$ as the phase. The decay rate for such oscillatory integrals was studied by the school of V. I. Arnold, which highlighted the importance of Newton polyhedra. Of particular importance for this thesis is an algorithm developed by A. N. Varchenko in the 1970s yielding a way to calculate the decay rate when $\phi$ is a two-dimensional real ...