函数空间实变理论及其上的算子性质(有界性及紧性)是调和分析的核心研究内容之一,并为数学和物理中许多问题的研究提供了重要的工作空间、工具和方法. 特别地, Hardy型空间和Sobolev空间是调和分析和偏微分方程研究的关键工具.本博士学位论文致力于研究各种Hardy型空间的实变特征及其在双线性分解和旋度散度引理等问题中的应用,以及Sobolev型空间的特征及其在Gagliardo--Nirenberg不等式的应用.

本学位论文主要分为三个方面.
首先, 研究了欧氏空间上局部Hardy空间中函数与其对偶空间中函数的点态乘积的双线性分解, 并证明此分解在某种意义上是最佳的(sharp). Bonami等人在[Ann. Inst. Fourier 2007]中提出了如下猜测: 是否存在欧氏空间上Hardy空间H¹（Rⁿ）中的函数与其对偶空BMO中函数的点态乘积的双线性分解？ 该猜测被Bonami等人于[J. Math. Pures Appl. (9) 97 (2012)]彻底解决. 并且, Hardy空间H^p（Rⁿ）(p€(0,1)]中的函数与其对偶空间中函数的点态乘积的双线性分解问题被Bonami等人于[J. Math. Pures Appl. (9) 131 (2019)] 彻底解决.本文彻底的解决了此问题在局部Hardy空间中的情形, 并将此结果应用于旋度散度引理的研究中. 这部分的主要创新点是巧妙的构造出了双线性分解中的目标空间的形式.其次, 系统地发展了与球拟Banach 函数空间相关的交换子理论以及(弱)Hardy型空间的实变理论, 并应用其思想解决了Cleanthous等人于[J. Geom. Anal. 27 (2017)] 所提出的关于混合范数Hardy空间的对偶空间的公开问题在欧氏空间上的情形. 对于这个对偶空间的构造,本文突破了经典Campanato-型空间只需体现一个原子特征的构造方法, 取而代之,使用有限个原子的特征来构造理想的对偶空间.最后, 通过首次引入了与球拟Banach 函数空间相关的Sobolev 空间, 用弱型空间与函数差分形式给出了与球拟Banach函数空间相关的Sobolev空间范数的等价刻画,并应用此特征刻画建立了相关的Gagliardo--Nirenberg不等式.这些结果推广了Brezis等人于[Proc. Natl. Acad. Sci. USA 118 (2021)]得到的结果.并且又将Brezis等人的结果推广到了齐型空间的情形.对于这两个推广, Brezis等人文章中的方法是不适用的,因为本文考虑的空间一般不具有平移不变性,从而无法使用极坐标性质.这部分主要创新点是通过充分利用Poincar\'e不等式, 外插的思想, 极大函数算子的精确算子范数, 以及差分与导数之间的集合包含关系,克服了空间缺乏平移不变和旋转不变性所带来的本质困难.这些结果丰富了调和分析中各种函数空间的实变理论,并为相关的分析问题的研究提供了更多的工作空间和理论方法.

The real-variable theory of function spaces and the properties of operators on them are one of the most important contents in harmonic analysis, and provide important workspaces, tools and methods for the study of many problems in mathematics and physics. This dissertation is devoted to the study of the real-variable characterizations of various Hardy-type spaces and their application to bilinear decomposition problems, as well as the real-variable characterizations of Sobolev-type spaces.

The main content of this dissertation is divided into three aspects.

First, this paper studies the bilinear decomposition of the point product of a function in local Hardy space and its dual space on Euclidean space, and proves that this decomposition is sharp in some sense. In [Ann. Inst. Fourier 2007], Bonami et al. proposed the following open problem: Is there a bilinear decomposition of the point product of a function in Hardy space and its dual space BMO on Euclidean space? This problem is completely solved by Bonami et al. [J. Math. Pures Appl. (9) 97 (2012)]. In [J. Math. Pures Appl. (9) 131 (2019)], a bilinear decomposition theorem for multiplications of elements in $H^p(\rn)$ and its dual space was established when $p\in(0, 1)$, and the sharpness of this bilinear decomposition was also obtained therein. This paper completely solves the problem in the case of local Hardy space, and applies this result to the study of curl divergence lemma. In order to solve this problem, this paper systematically develops the real-variable theory of Orlicz-slice type space on Euclidean space. Secondly, this paper systematically develops compactness characterizations of commutators on the ball Banach function space and the real-variable theory of (weak) Hardy-type spaces associated with the ball Banachfunction space. As an application, it solves the problem of the dual space of mixed-norm Hardy space proposed by G. Cleanthous, AG Georgiadis and M. Nielsen in [J. Geom. Anal. 27 (2017)] on Euclidean space. Finally, this paper introduces the Sobolev space related to the ball Banach function space for the first time, and gives the equivalent characterization of the Sobolev space norm related to the ball Banach function space in terms of the weak Lebesgue space and the function difference. The related fractional-order Sobolev and Gagliardo--Nirenberg inequalities are established by applying this characterization. These results generalize the results obtained by H. Brezis, J. Van Schaftingen and P.-L. Yung in [Proc. Natl. Acad. Sci. USA 118 (2021)]. Further, this paper extends the results of H. Brezis, J. Van Schaftingen and P.-L. Yung to the case of homogeneous spaces.