关于单边积分算子的新实变理论

Many important issues in Mathematics and Physics can often be attributed to the boundedness of operators on some function spaces, while this boundedness can not be characterized without the real variable theory of the corresponding function spaces. The one-sided integral operators in this subject come from the study of ergodic theory, and they are closely related to nonlinear dynamics and fractals, thus the real variable theory of integral operators has been one of the important components of harmonic analysis. The applicant and his collaborators have obtained a series of important results on integral operators. In particular, they got the weighted weak type estimates of one-sided oscillatory integrals. Also they have introduced a new class of one-sided Triebel-Lizorkin function spaces and established the weighted boundedness of one-sided C-Z integral operators on these spaces. This project will be devoted to the further research on the real variable theory of one-sided integral operators and their related function spaces, including establishing a criteria to determine the inner relation between the weighted boundedness of one-sided oscillatory integrals and one-sided singular integral operators; introducing the one-sided multilinear Reimann-Liouville operators and establishing their boundedness; establishing some real-variable characterizations of some one-sided function spaces such as one-sided Morrey space and Besov space; extending the application scope of the two-dimensional one-sided maximal operators(weight functions).

数学和物理中的许多重要问题均可归结为算子在某些函数空间的有界性，而刻画算子的有界性离不开相应函数空间的实变理论；单边积分算子源自遍历理论的研究且与非线性动力学及分形学密切相关，因此相关于单边积分算子的实变理论一直是调和分析的重要内容之一. 申请人及合作者已获得一系列关于积分算子有界性的重要结果；特别地，建立了单边振荡积分算子的加权弱型估计，引进了一类单边加权Triebel-Lizorkin空间并获得了单边C-Z奇异积分算子在其上的有界性. 本课题拟进一步发展单边积分算子及其相关函数空间的实变理论，其中包括建立单边奇异积分与单边振荡积分加权有界性的一类判定准则；完善单边分数次积分有界性理论；引进并研究相关于单边多线性Reimann-Liouville算子的函数空间并获得其有界性；建立包括单边Morrey空间和Besov空间在内的函数空间实变理论；拓广二维单边极大算子(权函数)性质的应用范围.

数学和物理中的许多重要问题均可归结为算子在某些函数空间的有界性，而刻画算子的有界性离不开相应函数空间的实变理论. 项目组研究了单边振荡积分的加权有界性质； 建立了单边振荡积分加权Lp有界性的判定准则，包括对应的极大算子的加权强型估计；引进了单边强型和弱型Morrey空间；并研究了单边Hardy-Littlewood极大算子、单边C-Z奇异积分算子和Riemann-Liouville分数次积分算子在单边Morrey空间上的加权有界性质；利用C-Z奇异积分算子交换子的有界性质刻画了Morrey空间；研究了单边振荡积分的从加权单边Hardy(H1)空间到加权L1的有界性质；研究了单边C-Z奇异积分算子、单边离散面基函数和Riemann-Liouville分数次积分算子的交换子在加权单边Triebel-Lizorkin空间上的加权有界性质；得到了由Riemann-Liouville分数阶的导数定义的Jeffrey流体方程的解析解，并通过图像表示出介质指标和分数阶指标变化引起的流体性态的变化. 针对非线性分数阶微分方程的Cauchy问题，研究了其在Morrey型空间中解的存在唯一性质. 另外，项目组分别在p进域上和海森堡群上的Hardy算子以及多线性Hardy算子的加权估计、多线性Littlewood-Paley算子的加权有界性、无穷区间上一类非线性积分边值问题的正解存在性、相关于交换算子的Hardy空间的分解、一类弱非线性方程解的爆炸率问题、Clifford分析中的算子理论和Helmholtz方程的边值问题、相关于n维实数空间中有界开Lipschitz子集的分数阶容量的性质、两类物种的趋化方程在齐次Besov空间上的存在性和Gevrey正则性等领域分别发表若干新结果.