Continuous And Discrete Fourier Restricted Estimation And Bounded Area Of Wave Equation And Its Related Mathematical Problems

The thesis consists of three chapters. In the first chapter, we consider a funda-mental problem lying in the core of modern harmonic analysis which is the Fourier restriction estimate for the measures supported on surfaces. Around this concern, we shall study Strichartz’s estimates in its linear and bilinear forms; bilinear restriction estimate; Carleman’s estimate and the unique continuation and finally the point-wise convergency of the Schrodinger evolution operator. Among them, quite a large number of open questions are still unsettled ranging from the remnant conjectures in classical theory to uncultivated fields which draw extremely few attentions. Chapter two devotes to the Schrodinger operator on compact manifolds, where we focus on the introduction of Bourgain’s pioneering work and his conjecture. Finally, we give emphasis to the mixed-Cauchy problem for a nonlinear system of wave equations posed on a compact domain. As for the Cauchy problems, there has been an active research in the past decades, flowering out scores of achievements. Especially in the central mathematical fields where harmonic analysis and functional analysis are mainly employed to study the existence and uniqueness of the global solutions, scattering theory, blow up phe-nomenon and their mechanism, many extraordinary accomplishments arose. However, in real applications we have to consider the system with boundary conditions. Thus it is vital and practical to study the mixed-problems. Limited by the strength of the method, the researches in correspondence are very few and still in its infancy. French mathematicians, represented by Lebeau, Burq and Planchon have obtained some mile stones in this direction.This article is the thesis of the author for a requirement of master’s degree, aiming at collecting some basic questions in harmonic analysis. The author discussed the Fourier restriction theorem using the smooth surface of fractional order as a model. Chapter two is a brief summary of the Schrodinger operator on compact manifolds which is restricted in a short paragraph. We believe that this direction will emerge in the future. For this reason, the author not only rephrase and interpret the work by Burq, Gerard and Bourgain etc. but also supplement their missing details and give; new proofs to the propositions therein whose original arguments are hard to get into. The third chapter is the author’s work published in Application Mathematicae. In addition, we insert some elementary frame work based on the language of semigroup theory. An appendix is in the end to exhibit an Euclidean version of bilinear eigenfunction estimates and estimate on Weyl sums. The former helps the reader understanding the way of using pseudo-differential operators while the latter indicates the connection of discrete restriction problem to additive theory of numbers.