Heat Equation And Harmonic Evolution Equation And Littlewood-Paley Theory For The Semigroups

In the last decades the theory of semigroups have been used successfully in the development of a theory of PDEs,one of the most famous results is the fractional Laplacian operator which was studied by Caffarelli and Silvestre.And now,the theory related with fractional operators becomes a hot topic in Harmonic Analysis and PDEs.The main aim of this thesis is to development the parabolic semigroup theory to deal with heat equation and harmonic oscillator evolution equation.Then,we studied the Littlewood-Paley g-functions of semigroups associated with these parabolic oper-ators,and oscillation operators of Poisson semigroups associated with these parabolic operators also considered.This thesis is consist of four chapters.In Chapter 1,we recalled the development of fractional operators,Littlewood-Paley theory,oscillation operators and some basic parabolic equations,and presented the motivation of the thesis,and stated the research methods and main results of the thesis.In Chapter 2,we first developed the parabolic semigroups theory.Then,we used the parabolic semigroups and parabolic vector-valued Calderon-Zygmund theory to study the heat equation(?)tu-△u=fand harmonic oscillator evolution equation(?)tu-△u+|x|2=fwe got the weighted mixed-norm Sobolev estimates of solutions associated with these equations.We also considered the corresponding Cauchy problems,proved a.e.conver-gence and weighted mixed-norm estimates for solutions of these equations above.In Chapter 3,we used the parabolic semigroups and parabolic vector-valued Calderon-Zygmund theory to consider the Littlewood-Paley g-functions of semigroups associated with parabolic operators(?)t-△ and(?)t-△+|x|2,proved the Lp inequalities of these g-functions.In Chapter 4,considered oscillation operators of Poisson semigroups associated with parabolic operators(?)t-△ and(?)t-△+|x|2;we proved the Lp boundedness of these oscillation operators by using the parabolic semigroups theory and vector-valued Calderon-Zygmund theory,the case L∞ of oscillation operators also considered.