The Riesz Average Of Fourier Restriction Estimates And The Klein-gordon-of Hartree Equations Are Then,

This dissertation is devoted to the Lp boundness of Bochner-Riesz mean operator associated with convex finite type hypersurface, the Fourier restriction problem and the low regularity of Klein-Gordon-Hartree equation. Both Bochner-Riesz and Fourier restriction conjectures are still open, a well-known big challenge, which have been at-tracting the focus of many established mathematicians. These analysis problems are connected to the study of partial differential equations, for instance, the Fourier re-striction estimates are associated with the Strichartz estimate which play an important role in studying the dispersive equations. With this view, we aim to partially solve or generalize both the conjectures in some special situations.In this preface, we introduce some tools and ideas from harmonic analysis, includ-ing stationary phase, oscillatory integral estimate, and the properties of orthogonality and cancellation. These are effectively and frequently used in this thesis.In the second chapter, we shall establish the global existence of Klein-Gordon-Hartree equation below the energy space. We shall make use of the Bourgain's Fourier truncation method to split the initial data into low frequency part and high frequency part. We then evolve the high frequency part with the original equation and invoke the global existence theory for small data to get a global solution; On the other hand, we shall evolve the low frequency part by the difference equation to obtain a local solution. It then needs to extend this local solution to be a global one. There are two essential difficulties in doing this. The first one is that the Strichartz estimate is not necessar-ily sharp to the case where the equation is subconformal. The second one arises in the nonlocal nonlinearity (|x|-γ*|φ|2)φ. The convolution term (viewed as a negative derivative) prevents us from using the refined Strichartz estimate. For the first diffi-culty, we take advantage of the flexibility of admissible pair of Klein-Gordon equations' Strichartz estimate. To overcome the second difficulty, we construct a commutator and establish this commutator estimate by exploiting cancellation property and utilizing Coifman-Meyer multilinear multiplier theorem.The third chapter is devoted to the Lp-boundness of Bochner-Riesz mean operator associated with convex finite type hypersurface. Bochner-Riesz conjecture is a famous open problem in harmonic analysis, we aim to generalize Lp bound of the Riesz means operator associated with the nonvanishing Gaussian curvature level set E to one asso-ciated with convex finite type hypersurface. To this end, we first recall the background of Bochner-Riesz conjecture and state our result,and then reduce our proof of the main theorem to Lp-estimate of a Hormander-type oscillatory integral operator(see Theorem 2.2.1 in [51]). These ideas are due to [48,51]. We shall utilize the TT* argument to reduce the proof to establish a variable oscillatory integral estimate. Finally, we employ the idea in [5] as well as the stationary phase argument to complete the proof. What we would like to mention is that authors in [24] obtained a better result by using different argument.In Chapter IV, we focus on the Fourier restriction problem. As well as Bochner-Riesz conjecture, the Fourier restriction conjecture is also a well-known open prob-lem. It is connected to many other conjectures, notably the Kakeya and the Bochner-Riesz conjecture. It is difficult and challenging to solve this problem, but we possibly solve them with some assumptions. For instance, Shao[54,55] proved that the cone and parab. restriction conjectures hold when the test function is in the cylindrically symmetric case, i.e., invariant under the spatial rotation. We observe that the norm Lqθ(Sn-1) of radial functions are equivalent. Inspired by this, we try to replace Ltq,x by Ltq(R;Lqrn-1drLθq(Sn-1)) (q≤q) to remove the radial assumption. If we could prove that (4.1.3) for q =q, then we would prove the conjecture; Unfortunately, we merely prove (4.1.3) holds for q≤2. This chapter includes four results. The first two are about the cone linear dual restriction estimate and the last two are on the parab.. In our first result, we remove the spatial radial assumption but add the hypothesis that test function's spherical harmonic expansion is finite. With this hypothesis, we needn't analyze the boundness of the Bessel function uniformly with its order k. In the second result, we effectively make use of the oscillation of eit|ξ| and stationary phase argument to analyze the asympotic behavior of the Bessel function, and thus remove the finite spherical harmonic expansion assumption. The last two results are about modified linear and bilinear parabolic restriction dual estimates. We also need the assumption that the test function's spherical harmonic expansion is finite in our proof. Many har-monic analysis tools such as spherical harmonic expand, TT* argument and Whitney decomposition are used in the proof. We hope that we could remove the finite spherical harmonic expand hypothesis in the future work.