Kakeya And Propagator Maximal Function Estimates And Nonlinear Schr(o|¨) Dinger Equations On Zoll Manifolds

The thesis consists of three chapters. In the first chapter, we consider a fundamen-tal problem lying in the core of modern harmonic analysis which is the Kakeya maximal function estimate. Based on combinatorics and an L2-estimate for auxiliary maximal functions, we use a new method instead of induction on scales to prove Wolff’s L5/2 esti-mate which is the best result so far. Chapter two devotes to the pointwise convergence problem of free Schrodinger solutions, where our strategy is inspired by Bourgain’s re-cent breakthrough in this direction. The crucial observation is the use of multilinear restriction theorem. Chapter three is about the uniform local well-posedness of nonlin-ear Schrodinger equation on Zoll manifolds. We establish a multilinear Strichartz esti-mate in Bourgain spaces based on the multilinear eigenfuncion estimate. The Cauchy problems on nonlinear Schrodinger equations have undergone an extensive study in the past decades, including the central problems of global well-posedness, scattering the-ory and blowup phenomena along with the blowup mechanism. In applications, one usually considers the inhomogeneous media, where the Euclidean space is replaced by manifolds and the variable coefficient Laplacian operator arises.