本学位论文致力于研究各向异性n维欧氏空间上相关于球拟Banach函数空间Hardy空间的实变理论及其应用. 长期以来, 如何把起源于调和分析的函数空间实变理论从经典欧氏空间Rn推广到各向异性欧氏空间上受到了广泛关注. 各向异性欧氏空间上的函数空间不仅被应用到小波分析, 而且在偏微分方程和几何分析等学科中也起着重要作用. 此外,Sawano等人在[Dissertationes Math. 525 (2017), 1–102]中引入了Rn上球拟Banach函数空间及相关的Hardy空间, 这为研究Hardy型空间的实变理论提供了一致的框架. 本文将调和分析中上述两方面的问题结合起来, 在各向异性Rn上发展了一套相关于球拟Banach函数空间的Hardy空间实变理论.
令A是一个n × n阶的扩张矩阵, X是Rn上的球拟Banach函数空间,支持Fefferman–Stein向量值极大不等式, 并且使得Hardy–Littlewood极大算子的幂在它的相关空间上有界. 本文首先通过非切向主极大函数引入了相关于A和X的Hardy空间HAX(Rn). 其次, 建立了HAX(Rn)的包括径向和非切向极大函数, (有限)原子和分子等在内的特征刻画. 最后, 通过建立线性算子在HAX(Rn)上的有界性准则, 得到了Calder′on–Zygmund算子从HAX(Rn)到X或到HAX(Rn)的有界性. 所有这些结果具有广泛应用. 特别地, 当被应用到Morrey空间和Orlicz-slice空间时, 部分结果也是新的. 由于X没有假设有绝对连续的拟范数, 这导致HAX(Rn)没有好的稠密子空间,因此Hardy空间原子分解的经典证明方法不再适用. 为克服此困难, 本文将HAX(Rn)中的分布与Schwartz函数做卷积, 先对此卷积做Calder′on–Zygmund分解, 再利用Banach–Alaoglu定理和级数的对角线法则得到了HAX(Rn)中分布的原子分解. 此外, 为了克服X没有范数的具体表达式所带来的本质困难, 本文采用了将X连续嵌入到加权Lebesgue空间并充分利用加权空间的已有结论来克服这一本质困难.
具体地, 本学位论文研究了以下内容.
令A是一个n × n阶的扩张矩阵, X是Rn上的球拟Banach函数空间满足上述提到的两个假设. 本文首先通过非切向主极大函数, 引入了相关于A和X的各向异性Hardy空间HAX(Rn). 利用几种极大函数之间点态可比较的性质以及Hardy–Littlewood极大算子的Fefferman–Stein向量值不等式, 建立了HAX(Rn)的径向和非切向极大函数刻画.
其次, 利用原子的Hardy–Littlewood极大函数在其支集外的点态估计, 向量值不等式, 以及Hardy–Littlewood极大算子的幂在X的相关空间上的有界性, 证明了原子刻画中的重构定理. 另一方面, 由于X没有假设有绝对连续的拟范数, 这导致HAX(Rn)没有好的稠密子空间,为此, 先对HAX(Rn)中分布与Schwartz函数的卷积做Calder′on–Zygmund分解. 再利用Banach–Alaoglu定理和级数的对角线法则得到了HAX(Rn)中分布的原子分解. 此外, 由于X的范数没有具体表达式, 该分解的收敛性证明的经典方法不再适用. 为此, 将X连续嵌入到加权Lebesgue空间并充分利用[Indiana Univ. Math. J. 57 (2008), 3065–3100]中加权空间的已知结论, 来证明原子分解在分布意义下成立.
最后,作为应用,得到了Calder′on–Zygmund算子T从HAX(Rn)到HAX(Rn)或到X的有界性.具体地, 在X有绝对连续拟范数的假设下, 建立了线性算子在HAX(Rn)上的有界性准则. 由此, 并利用X连续嵌入到加权Lebesgue空间以及Lebesgue控制收敛定理, 进一步去掉了这个额外假设, 从而获得了当X没有绝对连续拟范数时T在HAX(Rn)上的有界性. 此外, 利用HAX(Rn)的原子分解以及如上的类似讨论,得到T从HAX(Rn)到X的有界性.

This dissertation is devoted to the study the real variable theory of the n-dimensional Euclidean anisotropic Hardy spaces associated with ball quasi-Banach function spaces and its application. From then on, how to extend the real variable theory of function space which originated from harmonic analysis from classical Euclidean space Rn to anisotropic Euclidean space has attracted much attention. Function Spaces on anisotropic Euclidean Spaces are not only used in wavelet analysis, but also play an important role in partial differential equations and geometric analysis. Moreover, Sawano et.al. in [Dissertationes Math. 525 (2017), 1–102] have introduced ball quasi-Banach function spaces on Rn and the related Hardy spaces, providing a consistent framework for studying the real variable theory of Hardy type spaces. In this paper, we combine the above two aspects of harmonic analysis and develop a real variable theory of anisotropic Hardy space associated with ball quasi-Banach function spaces on Rn. 
Let A be a general n × n expansive matrix and X a ball quasi-Banach function space on Rn, which supports both a Fefferman–Stein vector-valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. This article first introduce the Hardy space HA X(Rn) associated with A and X, via the non-tangential grand maximal function. And then build some characterization of it, respectively, in terms of radial (grand) maximal functions, atoms or finite atoms, and molecules. Finally, obtain the boundedness of Calder′on– Zygmund operators from HA X(Rn) to X or to HA X(Rn) via first establishing a boundedness criterion of linear operators on HA X(Rn). All these results have a wide range of generality and, particularly, even when they are applied to the Morrey space and the Orlicz-slice space, the obtained results are also new. Since X does not assume an absolutely continuous quasi norm, this leads to HA X(Rn) having no good dense subspaces, so the classical proof of atomic decomposition in Hardy Spaces is no longer applicable. In order to overcome this difficulty, this article make a convolution with the distribution in HA X(Rn) and Schwartz functions, first use Calder′on–Zygmund decomposition to the convolution, and then obtained the atomic decomposition of the distribution in HA X(Rn) by Banach–Alaoglu theorem and the series diagonal rule. Moreover, in order to overcome the essential difficulty caused by X have no specific norm expression, this article use the X is continuously embedded into the weighted Lebesgue space and take full advantage of the known results of the weighted space to overcome this essential difficult. To be precise, this dissertation studies the following contents. Let A be a general n × n expansive matrix and X a ball quasi-Banach function space on Rn that satisfies the assumptions mentioned above. This article first introduced anisotropic Hardy space associated with A and X, via nontangential grand maximal function. By using the pointwise comparability of several maximal functions and the vector value inequality of Hardy–Littlewood maximal operator, the radial and non-tangential maximal function characterization of HA X(Rn) is established. Second, the reconstruction theorem of atomic characterization is proved by using the pointstate estimation of Hardy–Littlewood maximal functions of atoms, vector inequalities, and the boundedness of Hardy–Littlewood maximal operators on related space of X. On the other hand, X does not assume an absolutely continuous quasi-norm, which makes HA X(Rn) have no good dense subspace, to this end, first perform the Calder′on–Zygmund decomposition to the convolution of the distribution in HA X(Rn) with the Schwartz function. Then use the Banach–Alaoglu theorem and diagonal rule for series, we obtain the decomposition of distributions in HA X(Rn). Moreover, since there is no specific norm expression for X, the classical method of proving the convergence of decomposition is no longer applicable. To this end, let X continuous embedded into weighted Lebesgue space and taking advantage the known atomic characterization of weighted Hardy space in [Indiana Univ. Math. J. 57 (2008), 3065-–3100], to prove the convergence of decompositions in sense of distribution. Finally, as applications, we obtain the boundedness of Calder′on–Zygmund operators T from HA X(Rn) to X or to HA X(Rn). To be precise, we first establish a boundedness criterion of linear operators on HA X(Rn) under the assumption that X has an absolutely continuous quasi-norm. From this, and use the embedding of Xinto the weighted Lebesgue space and the Lebesgue dominated convergence, that extra assumption has been removed, which implies the boundedness of T on HA X(Rn) when X have no absolutely continuous quasi norm. Moreover, the boundedness of T from HA X(Rn) to X is obtained using the atomic decomposition of HA X(Rn) and similar discussions above.