| 中文题名： | 带矩阵权Triebel--Lizorkin空间的某些实变特征及其应用 |
| 姓名： | |
| 保密级别： | 公开 |
| 论文语种： | 英文 |
| 学科代码： | 070101 |
| 学科专业： | |
| 学生类型： | 硕士 |
| 学位： | 理学硕士 |
| 学位类型： | |
| 学位年度： | 2021 |
| 校区： | |
| 学院： | |
| 研究方向： | 函数空间及其应用 |
| 第一导师姓名： | |
| 第一导师单位： | |
| 提交日期： | 2021-06-12 |
| 答辩日期： | 2021-05-18 |
| 外文题名： | Some Real-Variable Characterizations and Their Applications of Matrix-Weighted Triebel--Lizorkin Spaces |
| 中文关键词： | 矩阵权 ; Triebel--Lizorkin空间 ; Littlewood--Paley函数 ; 球平均 ; Fourier 乘子 ; 对偶刻画 |
| 外文关键词： | matrix-weight ; Triebel--Lizorkin spaces ; Littlewood--Paley function ; ball-average characterization ; Fourier multiplier ; duality |
| 中文摘要： |
众所周知，经典Triebel-Lizorkin空间是函数空间实变理论的重要研究对象，并在调和分析及偏微分方程的研究中有着广泛应用．而为解决在研究多变元平稳随机过程及Toeplitz算子性质的过程中所遇到的问题, S. Treil及A. Volberg首次引入了矩阵权．有关矩阵权问题的研究不仅推动了调和分析中关于向量值函数空间实变理论的研究,还在概率论、信息控制论等其他数学领域的许多问题研究中起着重要的作用.本硕士学位论文研究了带矩阵权Triebel-Lizorkin空间的各种实变特征刻画, 并给出了其在Fourier乘子有界性研究中的应用.
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本学位论文的主要内容包括四个方面．第一，获得了带矩阵权Triebel-Lizorkin空间的Littlewood-Paley特征刻画，包括其上的Pectre极大函数特征，Lusin面积函数特征及或gλ*-函数特征．由于带矩阵权的极大函数向量值不等式是否成立仍有待研究,故难以通过近年来己有的刻画Triebel-Lizorkin型空间中的Littlewood-Paley函数特征刻画的方法证明本文所研究的空间中的相关特征刻画．为此，本文先巧妙地得到了带相关于矩阵权的约化算子Triebel-Lizorkin空间上的Littlewood-Paley函数特征刻画，再通过证明相关于矩阵权的范数与相关于该矩阵权的约化算子范数的等价性得到了带矩阵权Triebel-Lizorkin空间的Littlewood-Paley特征刻画．第二，获得了带矩阵权Triebel-Lizorkin空间在某些指标下的球平均函数刻画．第三，作为应用，证明了带矩阵权Triebel-Lizorkin空间中Fourier乘子的有界性.第四，通过证明带相关于矩阵权的约化算子的Triebel-Lizorkin空间的对偶空间得到了带矩阵权Triebel-Lizorkin空间的对偶空间刻画.所有这些结果的一个突出的创新性在于充分利用了带相关于矩阵权的约化算子的Triebel-Lizorkin空间上的各类特征刻画及其与带矩阵权空间的等价性，从而逃脱了己有的(部分)相应结果对带矩阵权的极大函数向量值不等式的依赖性.因此,本文结果极大地丰富了带矩阵权'Triebel-Lizorkin空间的实变理论,并为研究带矩阵权'Triebel-Lizorkin空间上算子有界性等其它分析问题提供了有利的工具. 具体地,本学位研究了以下内容. 第一,得到了α ∈R, p ∈(0, ∞), q ∈(0, ∞]，以及矩阵权W满足Ap(Rn,Cm)条件时的带矩阵权'Triebel-Lizorkin空间上的Peetre极大函数特征，Lusin面积函数特征，gλ*-函数特征.由于带矩阵权极大函数向量值不等式仍是未知的，故T. Ullrich在刻画Triebel-Lizorkin空间中的Peetre极大函数特征，Lusin面积函数特征，gλ*-函数特征时所采用的方法不再适用．本文采用了一种类似于M.Frazier和S. Roudenko在刻画带矩阵权Triebel-Lizorkin空间中g-函数特征时使用的证明方法并对其进行了修改.通过引入相关于矩阵权的约化算子的Peetre极大函数,Lusin面积函数以及gλ*-函数,并证明其与带矩阵权的Peetre极大函数,Lusin面积函数以及gλ*-函数的等价性以及带矩阵权Triebel-Lizorkin空间范数的等价性得到了带矩阵权Triebel-Lizorkin空间的这些Littlewood-Paley函数特征刻画. 第二，得到了α∈ (0,2)，p ∈ (1, ∞)，q ∈[ p, ∞]以及矩阵权W满足Ap(Rn,Cm)条件时带矩阵权Triebel-Lizorkin空间的球平均函数刻画．需要指出的是，其上的球平均函数在q ∈ (1, ∞]都是可以被带矩阵权Triebel-Lizorkin空间范数控制的.但限于矩阵权的性质及带矩阵权Triebel-Lizorkin空间中函数的傅里叶变换并无紧支集的性质,在证明另一侧的控制关系时,本文仅能得到q ∈[p, ∞]时的结论的成立性. 同时,由于带矩阵权的极大函数向量值不等式仍是未知的, 故B.Li，M.Bownik,D. Yang和W.Yuan在证明带标量权Triebel-Lizorkin空间中的球平均函数刻画时所采用的方法不再适用. 本文通过证明相关于矩阵权的约化算子Triebel-Lizorkin空间中的球平均函数刻画及其于带矩阵权Triebel-Lizorkin空间中的范数的等价性,最终得到了带矩阵权Triebel-Lizorkin空间中的球平均函数刻画. 第三, 作为应用, 通过带矩阵权Triebel-Lizorkin空间中的Peetre极大函数特征刻画, 得到了在α ∈ R, p ∈ (0, ∞)，q ∈ (0, ∞]及矩阵权W满足Ap(Rn,Cm)条件时，Fourier乘子在Hormander条件假定下的有界性. 最后，给出了带矩阵权Triebel-Lizorkin空间在α∈ R, p ∈ (1, ∞)，q ∈ (0,∞]及矩阵权W满足Ap(Rn,Cm)条件下的对偶空间刻画. |
| 外文摘要： |
lt is well known that the classical Triebel-Lizorkin spaces are important research objects of real variable theory in function space, and they are widely used in harmonic analysis and partial differential equations. In order to solve the problems encountered in the study of multivariate stationary random process and the properties of Toeplitz operators, Treil and Volberg introduced matrix weights. The introduction of matrix weights not only promotes the study of the properties of high-dimensional weighted spaces in harmonic analysis, but also plays an important role in the study of physical informatics and otherfields. In this dissertation，we establish the characterization of various characterizations of real variable features and give its application in the study of Fourier operator.
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The main content of this dissertation includes four aspects. First,we obtain the Littlewood-Paley feature characterization of the matrix-weighted Triebel-Lizorkin spaces, including the Peetre maximal function feature, Lusin area function feature, and gλ*-function feature. Since there is no matrix-weighted maximal function vector-value inequality so far, it is difficult to prove the relevant Littlewood-Paley characterizations in the matrix-weighted Triebel-Lizorkin spaces studied in this paper by the methods of characterizing the Littlewood-Paley characterizations in Triebel-Lizorkin type spaces which have been used in recent years. In this paper, we first get the characterization of the reducing operator weighted Triebel-Lizorkin spaces. Then by proving that the equivalence of the norm related to matrix weight and norm related to its reducing operators, we obtain the Littlewood-Paley characterizations of the matrix-weighted Triebel-Lizorkin spaces. Second, we obtain the ball-average characterization of the matrix-weighted Triebel-Lizorkinspaces under some indices. Thirdly, as an application, we prove the boundedness of Fourier multiplier on the matrix-weighted Triebel-Lizorkin spaces. Fourthly, the dual space characterization of the matrix-weighted Triebel-Lizorkin spaces is obtained by proving the dual space of reducing operator weighted Triebel-Lizorkin spaces. An outstanding innovationof all these results is that we make full use of all kinds of characterizations on the reducing operator weighted Triebel-Lizorkin spaces and the relationship between the matrix weightand its reducing operators, so as to escape the dependence of the existing corresponding results on the matrix-weighted maximum function vector-value inequality. Therefore, the results in this paper greatly enrich the theory of the real transformation of the matrix-weighted Triebel-Lizorkin spaces and provide a useful tool for studying the boundedness of operators on matrix weighted Triebel-Lizorkin spaces and other analytical problems. Precisely, the remainder of this article is organized as follows. First,we obtain the Littlewood-Paley characterizations of the matrix-weighted Triebel-Lizorkin spaces including Peetre maximal function feature，Lusin area function feature, and gλ*-function feature when α∈ R, p ∈ (0, ∞) , q ∈ (0, ∞], and W∈Ap(Rn,Cm). The methods used in [36,2.8] is no longer suitable for describing such characterizations on the matrix-weighted Triebel-Lizorkin spaces since there is no matrix-weighted maximal function vector-value inequality. This paper uses a proof similar to the g-function feature characterization on the matrix-weighted Triebel-Lizorlin spaces in [13,1.1] and modifies it. By introducing the Peetre maximal function, Lusin area function and gλ*-function with the reducing operator relating to the matrix weight，and proving that they are equivalent to the Peetre maximal function，Lusin area function，and gλ*-function relating tothe matrix weight respectively,we obtain the Littlewood-Paley characterizations of thematrix-weighted Triebel-Lizorkin spaces. Second, we obtain the ball-average characterizations of the matrix-weighted TriebelLizorkin spaces when α∈(0,2), p ∈ (1,∞), q ∈ [p, ∞] , and W ∈Ap(Rn,Cm). Indeed, for any , there exists an  on its related ball-average spaces when q ∈ (1, ∞). However, due to the property of matrix weights and the fact that the functions on the matrix-weighted Triebel-Lizorkin spaces do not have compact supports, we can only obtain the otherside results in the case of q e [p, ∞]. Meanwhile, the methods of characterizingthe ball-average characterization on the scale weighted Triebel-Lizorkin spaces used in [21, Theorem 1.5] is no longer feasible. In this paper，we used the reducing operators to overcome the obstacle.Third，as an application，we obtain the boundedness of Fourier multiplier on thematrix-weighted Triebel-Lizorkin spaces in the assumption of α∈ R, p ∈ (0, ∞)，q ∈ (0, ∞], W ∈Ap(Rn,Cm)， and the Hormander condition with the help of the Pee-tre maximal function. Finally, we obtain the duality of the matrix-weighted Triebel-Lizorkin spaces when α∈ R,p ∈ (1, ∞)，q ∈ (0,∞] , and W ∈Ap(Rn,Cm) with the help of the reducing operators. |
| 参考文献总数： | 44 |
| 馆藏号： | 硕070101/21026 |
| 开放日期： | 2022-06-12 |
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