双线性分解和容量及其在算子有界性中的应用

It is known that establishing boundedness of various operators on the function spaces by virtue of their real variable theory has been the fundamental topic in harmonic analysis. This is because many important problems in physics and mathematics can be deduced to proving the boundedness of certain operators on the function spaces. The applicant and her collaborators have achieved a series of profound results on the real variable theory of function spaces, including the Hardy spaces, on various underlying spaces, as well on boundedness of singular integral operators. Based on the existing work of the applicant and her collaborators, this project will further establish the bilinear decomposition over the Euclidean space of elements from the Hardy space Hp and its dual space under the case 0<p<1 by using wavelet bases, with applications to the endpoint estimations of the div-curl lemma and commutators, so that to thoroughly solve the bilinear decomposition problem; develop a corresponding bilinear decomposition theory over the space of homogeneous type by using Auscher et al.’s wavelet theory; establish a bilinear decomposition theory of the Hardy space associated to operators and its dual space; completely solve the tracing law problem of the Riesz potential and its various deformations; use the capacity theory to describe the properties of the solutions of the heat equations over the setting of the space of homogeneous type; establish a capacity theory on the graph and use it to study the Sobolev trace inequalities.

应用函数空间实变理论研究算子在其上的有界性一直是调和分析核心内容之一，这是因为数学和物理中的许多重要问题均可归结为算子在函数空间的有界性. 申请人及其合作者在包括各种底空间上的Hardy空间在内的函数空间实变理论和奇异积分等算子有界性方面获得了一系列深刻结果. 本项目拟在申请人及其合作者已有工作的基础上，利用小波基，进一步建立欧氏空间上当0<p<1时Hardy空间Hp与其对偶空间中元素乘积的双线性分解，并应用于散度-旋度引理及交换子有界性的端点估计中，从而彻底地解决其双线性分解问题；应用Auscher等人新进所建立的齐型空间上的小波理论，建立上述双线性分解在齐型空间上的相应理论；并发展相关于算子的Hardy空间及其对偶空间的双线性分解理论; 建立完整的Riesz位势迹定理及其各种变形；利用容量刻画齐型空间上热方程解的性质；系统建立图上的容量理论并用以研究图上的Sobolev迹不等式等问题.

函数空间实变理论与算子有界性理论是现代调和分析的核心研究内容之一，在偏微分方程、位势论和几何分析等学科中均有着重要应用. 本项目在度量空间上的Hardy空间实变特征刻画、欧氏空间和度量空间上的Hardy空间与其对偶空间的双线性分解理论及其在算子有界性中的应用、Riesz-(Hardy-)Morrey位势的限制性准则及其在某些分数次偏微分方程解的Hölder-Zygmund正则性中的应用、Gauss测度空间上的Sobolev型容量理论、用泛函-分析的方法研究对数Sobolev空间的容量以及相应的对数型周长和对数型平均曲率等问题、利用度量空间上的Sobolev容量刻画带Newton-Sobolev数据的热方程解的限制性质、与分数次梯度算子相关的Hardy-Sobolev空间及其容量理论以解决Bourgain-Brezis问题并应用于Fefferman-Stein的BMO分解、度量空间上的热核与Hardy不等式等方面均获得了一系列原创性的研究成果和重要进展.上述成果为研究调和分析与偏微分方程等学科中的相关分析问题提供了新的工作空间和方法.这些成果已发表于19篇学术论文和1本Springer出版社《Lecture Notes in Math.》系列专著.