平均算子族和截断奇异积分算子族的加权变差不等式

The variation inequalities for family of operators have been the hot subjects of probability, ergodic theory and harmonic analysis, they play an important role in the pointwise convergence of the underlying family of operators. The variation inequalities for average operators and truncated singular integrals have been well studied, but there are few results about the weighted variation inequalities for average operators and truncated singular integrals with rough kernels. Further, the proof of almost known weighted variation inequalities depend on Sharp maximal function estimate, which is not a powerful tool. Applicant has got the variation inequalities for average operators and truncated singular integrals with rough kernels by Fourier transform estimates, Littlewood-Paley theory and interpolation theorem. Based on his early works, applicant will establish the weighted variation inequalities for average operators and truncated singular integrals with rough kernels by using Fourier transform estimates, Littlewood-Paley theory and interpolation theorem with change of measure in this project, and give a general way to show weighted variation inequalities by using related Fourier transform estimates. This project will produce a new way for related problems in probability theory and ergodic theory, it also will supplement and complement singular integrals theory.

算子族的变差不等式一直是概率论、遍历理论和调和分析中的热点问题，它在算子族的点态收敛问题中有重要应用。平均算子族和截断奇异积分算子族的变差不等式已有众多结果，但带粗糙核的平均算子族和截断奇异积分算子族的加权变差不等式还鲜有研究。另外，现有的加权变差不等式的证明均依赖于Sharp 极大函数估计，有一定的局限性。申请人前期已运用Fourier变换估计、Littlewood-Paley理论和算子插值定理证明了带粗糙核的平均算子族和截断奇异积分算子族的变差不等式。基于其前期工作，申请人将在本项目中运用Fourier变换估计、Littlewood-Paley理论和变测度算子插值定理探讨带粗糙核的平均算子族和截断奇异积分算子族的加权变差不等式，并给出运用Fourier变换估计证明加权变差不等式的一般方法。本课题将为概率论和遍历理论的相关研究提供新思路，也将丰富和完善奇异积分理论。

算子族的变差不等式一直是概率论、遍历理论和调和分析中的热点问题，它在算子族的点态收敛问题中有重要应用。平均算子族和截断奇异积分算子族的变差不等式已有众多结果，但带粗糙核的平均算子族和截断奇异积分算子族的加权变差不等式还鲜有研究。另外，现有的加权变差不等式的证明均依赖于Sharp 极大函数估计，有一定的局限性。本课题组运用Fourier变换估计、Littlewood-Paley理论和变测度算子插值定理探讨带粗糙核的平均算子族和截断奇异积分算子族的加权变差不等式，并给出运用Fourier变换估计证明加权变差不等式的一般方法。具体地， 课题组可得了以下结果：. (1).带粗糙核的截断奇异积分和极大算子族的变差算子和跳算子在Lp空间上的有界性；.(2).带粗糙核的截断奇异积分和极大算子族的变差算子和跳算子在加权Lp空间上的有界性；.(3).恒等逼近算子族的变差算子、跳算子和振荡算子在Hardy空间上的有界性；.(4).带粗糙核的截断奇异Radon变换和极大算子族的变差算子在Lp空间上的有界性；. (5).向量值的沿曲线（面）的奇异积分在Lp空间上的有界性..本项目为概率论和遍历理论的相关研究提供新思路，也进一步丰富和完善奇异积分理论。