Boundedness Of Unimodular Multipliers On Function Spaces

This PH.D thesis focus on the behavior of unimodular Fourier multipliers on Besov type spaces, including modulation spaces, α-modulation spaces and Besov spaces. In addition, we try to answer the longstanding question associated with modulation spaces, a-modulation spaces and Besov spaces, that is, can we reformulate a-modulation spaces by interpolations between modulation spaces and Besov spaces?The modulation spaces has a long history, which was introduced by Fe-ichtinger by the short-time Fourier transform. It was used to characterize the time frequency phenomenon different from the dyadic decomposition of frequen-cy. The definition of modulation space was proved to be equivalent to a function space of Besov type, which ultimately initiated the study of modulation space in the field of harmonic analysis. In addition, with the help of uniform decom-position, the boundedness of dispersive semigroup can be obtained, which impel the research of dispersive equations on modulation spaces. Afterwards, Grober introduce the a-modulation space, which can be viewed as the intermediate space between modulation space and inhomogeneous Besov space. Intuitively speaking, modulation space and Besov space are just the two endpoints of a.We mainly consider on the difference and connection among these three types of function spaces, where the unimodular Fourier multipliers would play an es-sential role in reflecting them. One may see that, the behavior of α-modulation space can just be the style like the intermediate function spaces between modu-lation spaces and Besov spaces. However, this intuition is wrong in some other cases, that is, the a can’t be simply viewed as the intermediate function spaces between modulation spaces and Besov spaces, we will use unimodular Fourier multipliers to reflect the former, and the complex interpolations to illustrate the latter.Firstly, we state the behavior of unimodular Fourier multipliers on mod-ulation spaces, α-modulation spaces and inhomogeneous Besov spaces. Under some derivative assumptions, we establish the boundedness of unimodular Fouri-er multipliers. By a improved assumption, we get the necessary conditions for the boundedness. On the assumption that more stronger, we get the sharp result. Our theorems generalize and enrich the previous results, including the result as-sociated with some dispersive semigroup. Meanwhile, our theorem can be used to characterize these three types of function spaces. Combined with the construc-tion of operator and the application of classical results of complex interpolation, the sharpness result, which can be regarded as a label of function spaces, can be used to deduce a negative answer of complex interpolation in some index range. Our theorem only need some restrictive conditions of potential index, which make us believe the correctness of full result.Then, we study the behavior of free solution of dispersive equations on mod-ulation spaces, α-modulation spaces and Besov spaces. On the premise of bound-edness, we get the convergence rate or divergence rate of the free solutions. We al-so use our methods to study the unimodular Fourier multipliers on homogeneous Besov spaces. Similarly, we get the sharp results under a derivative assumption including some dispersive semigroup. Moreover, we get the precise estimates of operator norms, and obtain the blowup rate near the singularity point. We now sketch the outline of this thesis by chapter.In Chapter1. we review the previous results of unimodular Fourier multipliers on modulation spaces, and show the main results in this paper.In chapter2, we introduce the definitions and equivalent characterizations of some function spaces. We show the basic properties and review some known theorems which will be used in this paper.In Chapter3, we make a systematic study of unimodular Fourier multipliers on modulation spaces, a-modulation sapces and inhomogeneous Besov spaces. We get the boundedness, necessary conditions of boundedness and the sharp con-ditions respectively, under the conditions of different intensity. In addition, we give a partial answer for the conjecture of complex interpolation on a-modulation space.In Chapter4, we still study the boundedness properties of the unimodular Fouri-er multiplier operator on a-modulation spaces and inhomogeneous Besov spaces. We improve the conditions for the boundedness of Fourier multipliers with com-pact supports. If μ is a radial function and satisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness of the corresponding unimodular Fourier multiplier operator on α-modulation spaces.In Chapter5, we study the convergence rate and divergence rate of free solutions of general dispersive equations. Our results indicate the independence between the rate and exact α.In chapter6, we focus on the behavior of unimodular Fourier multipliers on ho-mogeneous Besov spaces, we get the precise estimates of operator norms.