本学位论文致力于研究n维欧氏空间R^n上一类新的Besov型和Triebel-Lizorkin型空间理论.我们首先给出一个反例说明了经典的二进Hausdorff容量并不一定是Choquet容量, 这否决了D. R. Adams [Lecture Notes in Math. 1302(1988), 115-124] 的一个被多次引用的命题. 为此, 我们引进了一类新的二进Hausdorff容量并证明了它们是Choquet容量. 利用这些新的二进Hausdorff容量, 我们引进了四类新的函数空间, 包括齐性和非齐性的Besov型空间和Triebel-Lizorkin型空间以及它们的前对偶空间: Besov-Hausdorff空间和Triebel-Lizorkin-Hausdorff型空间. 这些新的空间将许多经典的函数空间统一在一个尺度下, 例如齐性和非齐性的Besov空间, Triebel-Lizorkin空间, Q空间, Hardy-Hausdorff空间, Morrey空间, Hardy-Morrey空间, Triebel-Lizorkin-Morrey空间以及某些Besov-Morrey空间和Campanoto空间. 从而, 我们回答了G. Dafni和J. Xiao在 [J. Funct. Anal. 208(2004), 377-422] 中提出的关于建立Q空间, Hardy-Hausdorff空间与Besov和Triebel-Lizorkin空间之间关系的公开问题. 此外, 我们研究了这几类空间的许多性质特征和等价刻画, 包括M. Frazier和B. Jawerth意义下的\vz变换特征, Sobolev型嵌入性质, 提升性质, 光滑原子和分子分解特征, 对偶性质, 小波分解特征, 差分特征, 振荡特征, 极大函数特征以及局部平均特征等. 作为应用, 我们得到了某些拟微分算子在这些空间上的有界性并建立了其上的迹定理. 特别地, 对非齐性Besov型空间和Triebel-Lizorkin型空间, 我们得到了 一些点态乘子和微分同胚复合算子在其上的有界性, 这为引进和研究上半平面和R^n中某些光滑区域上的Besov型和Triebel-Lizorkin型空间提供了理论基础. 值得指出的是, 由于Besov-Hausdorff和Triebel-Lizorkin-Hausdorff空间是通过Hausdorff容量定义的, 研究这两类空间经常需要建立一些非常精细的覆盖引理、并充分利用底空间的几何性质, 这是与研究经典的Besov空间和Triebel-Lizorkin空间的本质不同之处所在.

This dissertation focus on a new function space theory of Besov-Triebel-Lizorkin-type spaces on n-dimensional Euclid spaces R^n. First we give a counterexample to show that the classical dyadic Hausdorff capacity on R^n is not always a capacity in the sense of Choquet, which veto a well-knownproposition of D. R. Adams [Lecture Notes in Math. 1302(1988), 115-124]. We then introduce a new class of dyadic Hausdorff capacity, and prove that they are Choquet capacities. Via these new dyadic Hausdorff capacities, we introduce and investigate four new function spaces, including Besov-type spaces, Triebel-Lizorkin-type spaces and their preduals: Besov-Hausdorff spaces and Triebel-Lizorkin-Hausdorff spaces of both homogeneous andinhomogeneous types. These new function spaces unify and generalize several classical function spaces including Besov spaces, Triebel-Lizorkin spaces, Q spaces, Hardy-Hausdorff spaces, Morrey spaces, Hardy-Morrey spaces, Triebel-Lizorkin-Morrey spaces and some Besov-Morrey spaces and Campanoto spaces. Thus, we answer an open question posed by G. Dafni and J. Xiao [J. Funct. Anal. 208(2004), 377-422] on establishing the connections among Q spaces, Hardy-Hausdorff spaces and Besov-Triebel-Lizorkin spaces. Varies of properties and characterizations of these spaces, such as the \vz-transform characterization in the sense of M. Frazier and B. Jawerth, Sobolev-type embedding, lifting properties, smooth atomicand molecular decomposition characterizations, dual properties, wavelet decomposition characterization, difference and oscillation characterizations, maximal function and local mean characterizations, are established. As applications of some characterizations mentioned above, we establish the boundedness of some pseudo-differential operators and the trace theorem on all of these spaces. In particular, for inhomogeneous Besov-type and Triebel-Lizorkin-type spaces, we obtain the boundedness of some pointwise multipliers and the diffeomorphism composite operators on these spaces, which are the basis of the study for Besov-type spaces and Triebel-Lizorkin-type spaces on the upper plane and some smooth domains in R^n. It is remarkable that sinceBesov-Hausdorff spaces and Triebel-Lizorkin-Hausdorff spaces are defined via Hausdorff capacities, in the study of these spaces, sometimes we need some very delicate cover lemmas and geometricproperties of R^n, which is essentially different from classic Besov and Triebel-Lizorkin spaces.