| Harmonic analysis, which has been developing for nearly two hundred years, is a perfect and complete branch in mathematics, and its methods and tools are used in almost any other branch of mathematics. Its application to partial differential equations is one of important application and many tools, such as the method of interpolation, maximal functions and potential theory, become the necessary tools in PDE. On one hand, for the application on the boundary problem of the second elliptic equations, we can see C. E. Kenig's [35] and the reference therein. On the other hand, the applications on the Cauchy problem of the evolution equations are based on the estimate of oscillatory integral and potential to get the space-time estimate and establish wellposedness of the nonlinear problem, where the key is to establish the nonlinear estimate of the nonlinear term of the equations. In the area, J. Ginibre, T. Cazenave, C. E. Kenig, G. Ponce, L. Vega and J. Bourgain and so on did many excellent work and we can see Miao's [50], J. Bourgain's [8] and the reference therein and a list in the homepage of T. Tao (http://www.math.ucla.edu/-tao) for reference.In this paper we prepare to consider several problems of the partial differential equations by the methods of harmonic analysis. It is divided into two parts. In the first part we consider the dispersive equations, which contain three chapters. In Chapter 1, we consider the Lp-Besov estimate of the general free dispersive equation; In Chapter 2, we consider the Cauchy problem of the coupled Schrodinger-KdV equation; in the third chapter, we consider the Cauchy problem of a shallow water equation. In the second part, which contains one chapter, we consider the degeneraterelaxed Dirichlet problem-Chapter 1 We firstly consider the general dispersive equationwhere D = j = 1,2,...,n, P(D) is denned by its symbol. One can see there is no global regularity of derivation. But there really exist some kind of regularity. One is the increasing of the index of integrability and the other one is the local regularity of derivation, which can not only used on the nonlinear dispersive equation, but also can improve the Carleson's conjecture. When n = 1, Carleson's conjecture has been improved completely, while n > 1 , it is open, although there are a lot of authors contribute to it.S. Fukuma and T. Muramatu in [25] discussed the Lp-Besov estimate of the solution, where P, see Theorem 3 and Theorem 4 in [25]. In the chapter we consider the norm estimate of the formal solutionof the dispersive equation(0.1.3)where n ≥ 2, Ω is radial.The purposes of the chapter are: (1).To get some maximal Besov estimate which differ from [25]; (2).To generalize Theorem 3 and Theorem 4 in [25]. The main results are:Theorem 0.1.1 If, define operator T byPrecisely, there exists a constant C > 0, suchTheorem 0.1.2 T is bounded from where. Precisely, there exists a constant C > 0, such that(0.1.6)Theorem 0.1.3 radial, and outside of a neighborhood of origin, , where a is multiindex, thenT is bounded from there exists a constant C = C(n, a,r) > 0, suchthatTheorem 0.1.4 Under the hypothesis of Theorem 0.1.3, if(0.1.7)T is bounded from B, Pricisely, there exists a constant C =C(n,a,r,p) > 0, such thatRemark 0.1.1 It is easy to verify that satisfy the condition of Theorem 0.1.3. Similarly, one can verifywhere , also satisfy the condition of Theorem 0.1.3.Chapter 2 An interactional phenomenon between long waves and short waves has been studied in various physical situations. The short wave is usually described by the Schrodinger type equation and the long wave is described by some sort of wave equation accompanied with a dispersive term. Kawahara et al [34] studied the coupled systemwhere a,b,cL,cs are real constants. When the resonance condition CL = CS = 0 holds, this equation is known as the coupled Schrodinger-KdV equation. In the chapter we consider the Cauchy problem of general coupled...
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