Mixed norm Fourier restriction estimates for surfaces in ℝ3 and applications to PDEs

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 analytic function. In this thesis we shall be concerned with obtaining mixed norm $L^p - L^2$ Fourier restriction estimates for surfaces in $\mathbb{R}^3$. More precisely, we want to determine for which $(p_1,p_3) \in [1,2]^2$ the operator $R$ can be extended to an $L^{p_3}_{x_3}(L^{p_1}_{(x_1,x_2)}) \to L^2(\rho \mathrm{d}S)$ bounded operator. Such mixed norm Lebesgue spaces are of great interest in PDE theory. In the first part of this thesis we prove an extension of a more recent result of I. A. Ikromov and D. Müller to the mixed norm case. Ikromov and Müller considered local $L^p - L^2$ Fourier restriction estimates, i.e., estimates where one takes $\rho$ to be supported in a sufficiently small neighbourhood of a given point, and obtained sharp results for a wide range of surfaces in $\mathbb{R}^3$, including all analytic ones. We shall restrict ourselves though to surfaces given as graphs of functions $\phi$ which are either in adapted coordinates, or in non-adapted coordinates and have the so called linear height strictly less than two. In our proof we build on the methods of Ikromov and Müller and combine them with a method from a paper of J. Ginibre and G. Velo useful for handling mixed norm estimates. In the second part of this thesis we prove a wide generalization of a recent result of A. Carbery, C. E. Kenig, and S. N. Ziesler, who considered the Fourier restriction problem with a mitigating factor for surfaces in $\mathbb{R}^3$ given as graphs of homogeneous two-dimensional polynomials. The estimates they considered were global, i.e., the function $\rho$ took the role of a weight function whose support extends to the whole surface $S$. Compared to the proof of Carbery, Kenig, and Ziesler, our approach is considerably more elementary, avoiding any use of algebraic topology and algebraic geometry, and it allows one to cover a considerably larger class of surfaces, namely, the class of all surfaces which can be given as graphs of mixed homogeneous functions. An important step in our work is a classification of singularities for such surfaces, after which we can apply the previously developed elaborate methods and ideas of Ikromov and Müller and of Ginibre and Velo. Finally, as an application of the above results we use the Christ-Kiselev lemma in order to obtain new Strichartz estimates for large classes of partial differential and pseudodifferential operators.



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