函数空间的实变理论与算子有界性是现代调和分析的核心研究内容之一，其为诸如偏微分方程和位势理论等其它分析学科提供了许多重要的工作空间和研究工具. 例如，有界平均振动空间BMO和经典John--Nirenberg空间被广泛地应用于调和分析和偏微分方程的研究中; 同时，双权有界性准则在算子的加权有界性估计中也扮演着重要角色. 本硕士学位论文致力于研究一类新的John--Nirenberg-Q空间及双权有界性准则. 一是基于经典的John--Nirenberg空间和Q空间，引入并发展了一类新型John--Nirenberg-Q空间并建立了该空间的多种实变特征刻画，为John--Nirenberg型空间的研究提供了新的研究思路和方法; 二是建立了一类新的双权有界性准则，可适用于包括加权Morrey空间在内的许多函数空间，具有广泛的一般性.

本学位论文主要分为两个方面.

首先，引入并研究了一类基于等规格方体的John--Nirenberg-Q空间. 这类新空间结合了经典的John--Nirenberg空间和Q空间，其为调和分析特别是算子有界性问题的研究提供了新的工作空间和研究方法. 其中John--Nirenberg空间作为函数空间实变理论的重要研究对象在近几年受到了广泛关注，该空间是在1961年被John和Nirenberg在 [Comm. Pure Appl. Math. 14 (1961)] 中引入的. 之后，在2015年，基于经典的BMO空间和John--Nirenberg空间，Bourgain等人于 [J. Eur. Math. Soc. (JEMS) 17 (2015)] 引入了一类新的BMO型空间，即B空间. 再之后，为了更深入研究上述B空间和John--Nirenberg空间的性质，Jia等人于 [Sci. China Math. 65 (2022)] 引入了等规格的John--Nirenberg空间，并于后续文章中证明了Calderon--Zygmund算子等一些在调和分析中十分重要的算子在该空间上的有界性. 另一方面，BMO空间自1961年被John和Nirenberg在 [Pure Appl. Math. 14 (1961)] 中引入后一直是函数空间实变理论的重要研究对象，其在调和分析和偏微分方程中都有着广泛的应用. 在2000年，Essen等人进一步将BMO空间推广，于 [Indiana Univ. Math. J. 49 (2000)] 中引入Q_α空间(简记为Q空间). 本学位论文结合上述等规格John--Nirenberg空间与Q空间的特点，引入了一种新的空间，即等规格John--Nirenberg-Q空间(简记为JNQ空间)，并给出了该空间与一些重要函数空间的关系，以及该空间的多种实变特征刻画和其上复合算子的一些性质. 具体地，这一方面的研究主要包括以下内容: 其一，阐明了JNQ空间与一些经典函数空间的关系，包括等规格John--Nirenberg空间，Q空间，以及(分数次)Sobolev空间. 为此，建立了一种积分形式与求和形式的范数等价关系. 这种等价关系具有一定的普适性，可以被用于更广泛的函数空间，对研究等规格John--Nirenberg空间上Calderon--Zygmund算子，分数次积分，以及Littlewood--Paley算子的有界性起到了关键作用. 进一步，证明了Q空间可视为JNQ空间的极限情形. 此外，在某些指标范围下，证明了(分数次)Sobolev空间可连续嵌入到JNQ空间. 综合以上结果，最终得到了JNQ空间关于上下指标的完全分类. 其二，建立了JNQ空间的平均振动特征刻画，利用二进方体族引进了JNQ空间的二进空间，并证明了该二进空间是JNQ空间与等规格John--Nirenberg空间的交集. 证明该结果的困难之处在于，二进JNQ空间的定义中要求选取的方体族是互不相交且规格相等的，这对方体族的伸缩和变换有着很大的限制. 为了克服此本质困难，我们充分利用了二进方体所反映的几何性质. 最后，作为应用，研究了JNQ空间上复合算子的有界性特征，证明了JNQ空间上的左复合算子有界当且仅当其复合函数属于Lipschitz空间.

本学位论文的另一方面集中在建立了一种新的双权有界性准则，并给出了其在一些经典算子有界性研究中的应用. 事实上，有关经典算子的双权范数不等式的研究向来备受关注，这是因为其不仅对研究算子的有界性本身有着重要的贡献，且在偏微分方程中有诸多应用. 本学位论文建立了一种非常一般的双权有界性准则，推广了双权函数研究中已有的经典结论. 与经典结论相比，本文所建立的双权有界性判别准则适用于更多函数空间，包括加权Lebesgue空间，加权Lorentz空间，(Lorentz--)Morrey空间，以及变指标Lebesgue空间，具有广泛的一般性. 需要指出的是，为了建立这个准则，需要解决的主要困难是利用Calderon--Zygmund分解等工具建立了一个一般的加权good-λ不等式. 作为应用，首先获得了Calderon--Zygmund算子，Littlewood--Paley g-函数，Lusin面积函数，Littlewood--Paley g*λ-函数，以及分数次积分算子在上述空间的(双权)有界性. 得到判断上述这些算子的双权有界性的一个统一准则是本文的一个创新的体现. 其次，得到了相关于具有复有界可测系数的二阶散度椭圆算子的Riesz变换，Littlewood--Paley g-函数和分数次积分的双权有界性估计.

综上，这些结果丰富了函数空间的实变理论，并为(应用)调和分析与偏微分方程等其它分析学科提供了新的工作空间与理论工具. 

Both the real-variable theory of function space and the boundedness of operators are main research topics of modern harmonic analysis, which provide some important working spaces and new research methods for partial differential equations, potential theory and many other analysis disciplines. For example, bounded mean oscillation spaces BMO and classical John--Nirenberg spaces are widely used in harmonic analysis and partial differential equations; also, the two-weight boundedness criterion plays an important role in the weighted boundedness estimation of operators. This dissertation combines the classical John--Nirenberg spaces and Q spaces, establishes and studies a new kind of these John--Nirenberg-Q spaces. We establish several real-variable characterizations of John--Nirenberg-Q spaces, which bring about new thinking and approach for the study of John--Nirenberg spaces; also, we establish a new two-weight boundedness criterion, which is a generalization of the existing classical conclusions of two-weight functions and can be applied to many function spaces such as weighted Morrey spaces.

This dissertation includes two parts.

To begin with, we introduce and study the real-variable characterizations of John--Nirenberg-Q spaces via congruent cubes. These new spaces combine both the classical John--Nirenberg spaces and Q_α spaces, and provide new working spaces and to bring some new inspiring research methods for harmonic analysis, especially in study of the boundedness of operators. The real-variable theory of function space and the boundedness of operators are one of the main research topic of modern harmonic analysis, which provide some important working spaces and new research methods for partial differential equations, potential theory and many other analysis disciplines. For example, bounded mean oscillation spaces BMO and classical John--Nirenberg spaces are widely used in harmonic analysis and partial differential equations. As is well known, John--Nirenberg spaces, as an important research topic of real-variable theory of function spaces, have brought widespread attention in recent years. They were first introduced by John and Nirenberg in [Comm. Pure Appl. Math. 14 (1961)], Then, in 2015, based on classical BMO spaces and John--Nirenberg spaces, Bourgain et al. introduced a new type of BMO spaces in [J. Eur. Math. Soc. (JEMS) 17 (2015)], namely B spaces. After that, in order to study the properties of B spaces and John--Nirenberg spaces more deeply, Jia et al. established the congruent John--Nirenberg spaces in [Sci. China Math. 65 (2022)], and proved the boundedness of some important operators on these spaces in the further study, such as the boundedness of Calderon--Zygmund operators. On the other hand, because of they widely applications in harmonic analysis and partial differential equations, BMO spaces have been an important topic of real-variable theory of function spaces since it was introduced by John and Nirenberg in [Pure Appl. Math. 14 (1961)]. Furthermore, in [Indiana Univ. Math. J. 49 (2000)], Essen et al. generalized BMO spaces and established Q_α spaces (for short, Q spaces). Combining the characterizations of congruent John--Nirenberg spaces and Q spaces, we introduce a new space, John--Nirenberg-Q spaces via congruent cubes(for short, JNQ spaces). We obtain the relationship between JNQ spaces and some other important function spaces. Also, kinds of real-variable characterizations of JNQ spaces and some properties of composition operators on JNQ spaces are obtained. These results can be described precisely in the following three aspects. First, we reveal the relations between congruent John--Nirenberg-Q spaces and some other classical function spaces, including congruent John--Nirenberg spaces, Q spaces and (fractional) Sobolev spaces. It should be mentioned that in the proof of the relation between JNQ spaces and congruent John--Nirenberg-Q spaces, a main and skillful step is to establish an equivalence between norms of the integral type and the summation type. This equivalence is general and can be used in a much wider range of function spaces, which is of great help to the study of boundedness of some famous operators, including Calderon--Zygmund operators, fractional integrals, and Littlewood--Paley operators on congruent John--Nirenberg spaces. Also, we obtain that Q spaces are the limit spaces of JNQ spaces as p to infinity. Furthermore, for some non-negative az, we prove that the (fractional) Sobolev space is continuously embedded into the JNQ spaces. Second, we characterize JNQ spaces via mean oscillations and introduce the dyadic counterparts of JNQ spaces, which are proved to be the intersection of JNQ spaces and congruent John--Nirenberg spaces. The difficulty in proving this result is the definition of dyadic counterparts of JNQ spaces requires the collection of cubes to be mutually disjoint and congruent. In order to overcome this obstacle, some subtle geometrical properties of dyadic cubes from are needed. Finally, as applications, we discuss the composition operators on JNQ spaces and prove the left composition operators are bounded on the JNQ spaces if and only if the corresponding function belongs to the Lipschitz space.

Then, we establish a new two-weight boundedness criterion, and give its applications in the study of boundedness of some classical operators. As is well known, the research about the two-weight norm inequality of some classical operators has attracted great attention, not only because it plays an irreplaceable role in the study of the boundedness of operators, but also because it resulted in many applications in partial differential equations. In this dissertation, we establish a general two-weight boundedness criterion, which is a generalization of the existing classical conclusions of two-weight functions. Compared with the classical conclusions, our two-weight boundedness criterion is more general and could be applied to many function spaces, including weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. It should be mentioned that the main difficult need to be solved for establishing criterion is to establish a general weighted good-lambda equality via Calderon--Zygmund operators. As applications, using the above two-weight boundedness criterion, we first apply it to some classical operators. To be specific, we obtain the (two-weight) boundedness of classical Calderon--Zygmund operators, Littlewood--Paley g-functions, Lusin area functions, Littlewood--Paley g*λ-functions, and fractional integral operators on above spaces. The innovation is to establish a a unified methods to obtain the two-weight boundedness of those operators. Then, we obtain the boundedness estimation of Riesz transform, Littlewood--Paley g-functions and fractional intergal integral operators associated with second-order divergence elliptic operators with complex bounded measurable coefficients on the above spaces.

To sum up, above results enrich the real-variable theory of function space, as well as provide a new workspace and some theoretical tools for other analysis disciplines, such as harmonic analysis and partial differential equations.