众所周知, 经典Hardy空间及其各种变形是函数空间实变理论的重要研究对象, 并在调和分析特别是算子有界性问题研究中有着广泛应用.
本硕士学位论文在各向异性欧氏空间上引入了局部Hardy--Lorentz空间, 建立了该空间的
多种实变特征刻画和实插值性质, 并给出了其在伪微分算子有界性研究中的应用.


本学位论文主要包括两个方面. 第一, 引进了各向异性局部Hardy--Lorentz空间$h_A^{p,q}(\rn)$并得到了$h_A^{p,q}(\rn)$的几种实变特征刻画:
首先利用局部非切向极大函数定义了$h_A^{p,q}(\rn)$, 并给出了$h_A^{p,q}(\rn)$的多种极大函数刻画; 利用通过各向异性欧氏空间上的局部非切向主极大函数建立的Calder\'{o}n--Zygmund分解
给出了$h_A^{p,q}(\rn)$的(有限)原子分解刻画; 最后借助已得到的原子分解刻画建立了$h_A^{p,q}(\rn)$的分子分解刻画.
第二, 利用实插值方法以及已有的Lorentz空间的插值结果获得了Hardy--Lorentz空间$h_A^{p,q}(\rn)$的实插值性质并用以给出零阶伪微分算子在其上的有界性. 在此过程中, 明确了$h_A^{p,q}(\rn)$与非局部的经典各向异性Hardy--Lorentz空间$H_A^{p,q}(\rn)$之间的关系.
这些结果首次在各向异性欧氏空间上发展了局部Hardy--Lorentz空间, 丰富了各向异性欧氏空间上的函数空间实变理论, 并为调和分析、偏微分方程和小波分析等分析学科提供了更多的工作空间和研究工具.


具体地, 本论文研究了以下内容.


令$p\in(0,\fz)$, $q\in(0,\fz]$, 并给定一个扩张矩阵$A$. 本论文首先通过局部非切向极大函数, 在相关于$A$的各向异性欧氏空间上引入了局部Hardy--Lorentz空间$h_A^{p,q}(\rn)$, 其当$p=q$时即为经典的各向异性局部Hardy空间$h_A^{p}(\rn)$. 随后利用局部的径向极大函数和主极大函数给出了这类 的多种极大函数等价特征刻画, 这给该空间的应用带来了更多选择与便利. 对各向异性三元组$(p,r,s)$($p\in(0,1]$, $r\in(1,\fz]$以及$s\in\mathbb{N}$, 其中
$s\geq\lfloor(1/p-1)\ln b/\ln\lambda_-\rfloor$), 本论文定义了各向异性局部$(p,r,s)$-原子, 并利用通过局部非切向极大函数建立的各向异性Calder\'{o}n--Zygmund分解以及Lorentz空间的性质得到了$h_A^{p,q}(\rn)$的原子分解刻画. 在此基础上, 通过证明$h_A^{p,q}(\rn)\cap L^r(\rn)$ ($r\in[1,\fz]$)和$h_A^{p,q}(\rn)\cap C_c^\fz(\rn)$ ($C_c^\fz(\rn)$为具有紧支集的光滑函数全体组成的函数空间)在$h_A^{p,q}(\rn)$中稠密, 进一步得到了$h_A^{p,q}(\rn)$的有限原子分解刻画以及分子分解刻画. 此外, 本论文证明了在实插值意义下, 局部Hardy--Lorentz空间$h^{p,q}_A(\rn)$是$h^{p_1,q_1}_A(\mathbb{R}^n)$和$h^{p_2,q_2}_A(\mathbb{R}^n)$的
实插值中间空间 (其中$0<p_1<p<p_2<\infty$, $q_1,\,q,\,q_2\in(0,\infty]$), 同时也是$h^{p,q_1}_A(\mathbb{R}^n)$和$h^{p,q_2}_A(\mathbb{R}^n)$ (其中$p\in(0,\infty)$, $0<q_1<q<q_2\le\infty$) 的实插值中间空间, 由此说明了局部Hardy--Lorentz空间关于实插值方法封闭. 进一步地,
本论文利用实插值方法证明了$h_A^{p,q}(\rn)$是各向异性局部Hardy空间的实插值中间空间. 最后, 作为应用, 本论文获得了某些伪微分算子在$h_A^{p,q}(\rn)$上的有界性, 即当符号函数属于零阶符号类$S_{1,0}^0(A)$时, 其对应的伪微分算子$T$在$h_A^{p,q}(\rn)$($p\in(0,1]$, $q\in(0,\fz]$, $p\le q$)上有界; 为此, 本论文先给出了伪微分算子$T$的光滑化算子$T_t$所对应的符号$\sigma_t(x, \xi)$和核函数$K_t(x, z)$的精确导数估计, 这也是伪微分算子的有界性证明中的关键.

It's well known that the classical Hardy spaces and their various deformation are important research objects of real variable
theory in function space, and they are widely used in harmonic analysis, especially in the study of the boundedness of operator.
In this dissertation, we introduce the local Hardy--Lorentz spaces on anisotropic Euclidean spaces, establish the characterization of various characterizations of real variable features and the properties of real interpolation, and give its application in the study of pseudo-differential operator.


This dissertation is divided into two parts. Firstly, we bring in anisotropic local Hardy--Lorentz spaces $h_A^{p,q}(\rn)$ and obtain several characterizations of real variable features of it:
first we define $h_A^{p,q}(\rn)$ via local non-tangential maximal function and obtain several maximal functions characterization of $h_A^{p,q}(\rn)$, then obtain the (finite) atomic characterization via Calder\'{o}n--Zygmund decomposition associated with local non-tangential grand maximal functions on anisotropic $\rn$,
and finally establish the molecular characterization with the help of the atomic characterization obtained. Secondly, we obtain the real interpolation properties and the boundedness of zero order pseudo-differential operators on $h_A^{p,q}(\rn)$ using real interpolation methods and the known interpolation results on Lorentz spaces.
In this process, we clear the relationship between $h_A^{p,q}(\rn)$ and anisotropic Hardy--Lorentz spaces $H_A^{p,q}(\rn)$. These results develop local Hardy--Lorentz spaces on anisotropic Euclidean spaces for the first time, enrich the function space real variable theory on anisotropic Euclidean spaces and provide harmonic analysis, partial equation and wavelet analysis with more work spaces and research tools.


Precisely, the main points of this dissertation are as flows.


Let $p\in(0,\fz)$, $q\in(0,\fz]$ and a dilation $A$ is given.
In this paper, we introduce a local Hardy--Lorentz space $h_A^{p,q}(\rn)$ on an anisotropic Euclidean space with respect to $A$ via a local non-tangential maximal function, and when $p=q$, it becomes the classical anisotropic local Hardy space $h_A^{p}(\rn)$.
Then, the equivalent characterizations of these kinds of maximal functions are given by using local radial maximal functions and grand maximal functions, which brings more choices and conveniences to the application of this space.
For anisotropic triples $(p,r,s)$($p\in(0,1]$, $r\in(1,\fz]$ and $s\in\mathbb{N}$, where $s\geq\lfloor(1/p-1)\ln b/\ln\lambda_-\rfloor$), we define anisotropic local $(p,r,s)$-atoms. On this basis, by proving that $h_A^{p,q}(\rn)\cap L^r(\rn)$ ($r\in[1,\fz]$) and $h_A^{p,q}(\rn)\cap C_c^\fz(\rn)$ ($C_c^\fz(\rn)$ (a function space composed of all smooth functions with compact support set) are dense in $h_A^{p,q}(\rn)$, we further obtain the characterization of finite atomic decomposition and molecular decomposition of $h_A^{p,q}(\rn)$.
In addition, we prove that $h^{p,q}_A(\rn)$
is an intermediate space between $h^{p_1,q_1}_A(\mathbb{R}^n)$
and $h^{p_2,q_2}_A(\mathbb{R}^n)$ with
$0<p_1<p<p_2<\infty$ and $q_1,\,q,\,q_2\in(0,\infty]$, and also
between $h^{p,q_1}_A(\mathbb{R}^n)$
and $h^{p,q_2}_A(\mathbb{R}^n)$ with
$p\in(0,\infty)$ and $0<q_1<q<q_2\le\infty$
in the sense of real interpolation, which shows that the local Hardy--Lorentz spaces are closed with respect to the real interpolation method.
We further obtain
that the anisotropic local Hardy--Lorentz space $h^{p,q}_A(\rn)$ serves as
a median space
between two anisotropic local Hardy spaces via the real method.
In the end, as an application, we study the boundedness of some pseudo-differential operators, namely,
when symbol function belong to zero order symbol class $S_{1,0}^0(A)$, the corresponding pseudo-differential operator $T$ is boundedness on $h_A^{p,q}(\rn)$($p\in(0,1]$, $q\in(0,\fz]$, $p\le q$);
to this end,
we first give the exact derivative estimates of the symbol $\sigma_t(x, \xi)$ and the kernel function $K_t(x, z)$ associated with the smoothing operator $T_t$ of pseudo-differential operator $T$, which is the key to the proof of the boundedness of pseudo-differential operator.