经典的(局部)Hardy空间理论和奇异积分理论在调和分析、复分析、偏微分方程、几何分析等众多数学领域中起着重要作用. Hyt?nen意义下的新的非齐型空间包含互不包含的Coifman-Weiss意义下的齐型空间和具有多项式增长的非倍测度欧氏空间，具有广泛的一般性. 申请人及其合作者已在上述新的非齐型空间上的函数空间实变理论和算子有界性的研究中取得一定的成果，特别地， 建立了Hardy空间Hp(0＜p≤1)的原子、分子分解特征. 本项目拟在原有工作的基础上，发展新的非齐型空间上的Hardy空间Hp(0＜p≤1)和局部Hardy空间hp(0＜p≤1)的实变理论，其中包括对偶空间理论、插值定理、各种极大函数特征、(次)线性算子在其上有界的有界性准则以及hp的原子分解特征等，并将其应用于Bergman型奇异积分算子、分数次积分、Marcinkiewicz积分及其交换子的有界性的研究中.

The classical theory of Hardy spaces, localized Hardy spaces and singular integrals plays an important role in various fields of analysis and partial differential equations. The new non-homogeneous spaces in the sense of Hyt?nen unifies two different types of spaces, namely, Euclidean spaces with a non-doubling measure satisfying a polynomial growth condition and spaces of homogeneous type in the sense of Coifman and Weiss. The applicant and his collaborators have already made some achievements on the real-variable theory of function spaces and the boundedness of operators on the new non-homogeneous spaces, especially establishing the atomic characterization and the molecular characterization of Hardy spaces Hp for 0＜p≤1. Based on the previous work, the present project aims to develop the real-variable theory of Hardy spaces Hp for 0＜p≤1 and localized Hardy spaces hp for 0＜p≤1 on the new non-homogeneous spaces, including the theories of duality, interpolation, various maximal functions characterizations, boundedness criterions of linear and sublinear operators and the atomic decomposition of localized Hardy spaces hp. As applications, this project will consider boundedness of singular integrals of Bergman type, fractional integrals, Marcinkiewicz integrals and commutators on the corresponding function spaces.