Jump And Variational Inequalities Of Some Operators In Harmonic Analysis

The main aim of this thesis is to study the boundedness of the jump and variational operators related to the hypersingular integral operators with rough kernel,the hypersingular integral operators with variable kernel,and the average operators with variable kernel on certain function spaces.The main innovations of this thesis can be summarized as follows:1.In contrast to the known jump and variational inequalities for rough singular integral operators,we study jump and variational inequalities for a larger class of singular integral operators whose size conditions are rougher for the kernel.In the process of proving this conclusion,it is necessary to rely on tools such as Taylor expansion theorem and atomic decomposition on Hardy spaces to overcome the difficulties caused by the more complicated operators and rougher kernel functions.2.Based on the study of the innovation point 1,the weighted case was considered,and a larger class of weighted jump and variational inequalities for singular integral operators was established.This study not only extends the weighted results of jump and variation for the family of classical singular integral operators with rough kernels,but also includes the weighted boundedness of the maximal hypersingular integral operators with rough kernels.3.Jump and variational inequalities for a series of operators associated with variable kernels were established.Unlike the studies of innovation points 1 and 2,these operators are not of convolution type.Due to the invalidation of the Fourier transform and Plancherel’s theorem in the proof,the classical method of spherical harmonic function expansion is first used to handle the variable kernel.Then,some new results are obtained by discussing the variation or jump operators of the family of truncation operators associated with spherical harmonic functions.The specific research content of this dissertation is as follows:In Chapter 2,by using the Taylor expansion theorem and the atomic decomposition of Hardy spaces,we investigate that when the kernel function Ω belongs to Hq(Sn-1)with q=n-1/n-1+α and satisfies certain cancellation conditions,the jump and variation operators for the family of truncated hypersingular integral operator TΩ,α={TΩ,α,ε}ε>0 are bounded from the homogeneous Sobolev space Lαp(Rn)to the Lebesgue space Lp(Rn),where α≥ 0,1<p<∞.In Chapter 3,based on the study of jump and variational inequalities for hypersingular integral operators in Chapter 2,a class of weighted cases is established.More precisely,for α≥ 0,the jump and variation operators for the family of truncated hypersingular integral operator with rough kernel TΩ,α={TΩ,α,ε｝ε>0 are bounded from the homogeneous weighted Sobolev space to the weighted Lebesgue space,where the kernel function Ω belongs to Lq(Sn-1)(q>1)and satisfies certain cancellation conditions.In Chapter 4,the variational inequalities of hypersingular integral operators with variable kernels are discussed.Firstly,for the variable kernel satisfies Ω∈ L∞(Rn)× Lq(Sn-1),(q>max{1,2(n-1)/n+2α})and the certain cancellation conditions,we obtain the(Lα2(Rn),L2(Rn))boundedness of the variation operators for the family of the truncated hypersingular integral.Subsequently,by strengthening the smoothness condition of the second variable in the variable kernel,the(Lαp(w),Lp(w))boundedness of the variation operator for the family of a class of truncated hypersingular integral with smooth variable kernel are obtained,where 1<p<∞,w ∈ Ap.Finally,a variational inequality is established in the Sobolev-Morrey space by using weighted results.In Chapter 5,the jump and variational inequalities of averaging operators with variable kernels are studied.Firstly,L2(Rn)boundedness of the jump and variation operators for the family of averaging operator with rough kernel are obtained,where the variable kernel satisfies the size condition Ω ∈ O L∞(Rn)×Lq(Sn-1)(q>2(n-1)/n).Secondly,we obtain that the jump and variation operators for the family of averaging operator with smooth variable kernel are bounded on Lp(w),where 1<p<∞,w ∈ Ap.Finally,we extend the weighted results to Morrey spaces.