Rearrangements of the Haar system which preserve $\BMO$

Postorder rearrangement operators

We investigate the rearrangement of the Haar system induced by the postorder on the set of dyadic intervals in $[0,1]$ with length greater than or equal to $2^{-N}$. By means of operator norms on $\text{BMO}_N$ we prove that the postorder has maximal distance to the usual lexicographic order.

消失平均振荡与重排的连续性

Vanishing Mean Oscillation and Continuity of Rearrangements

本文研究了VMO中函数的递减重排，证明了映射f->f*保持消失平均振荡，并分析了BMO上映射的连续性条件，同时给出了对称递减重排的结论及多个示例。

准度量空间上的Haar基，以及同质类型乘积空间上函数空间的二进制结构定理

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

本文研究了在几何上双重的准度量空间中，利用二进制立方体系统显式构造Haar函数，并证明这些Haar函数构成L^p基。接着，针对同质类型空间X，利用这些Haar函数定义离散平方函数，从而定义H^1(X)和BMO(X)的二进制版本。在同质类型乘积空间X~上，证明BMO(X~)可以表示为有限多个二进制BMO空间的交集，同样适用于A_p(X~)、反Hölder权和双重权。此外，证明Hardy空间H^1(X~)是有限多个二进制Hardy空间之和，且X~上的强极大函数与有限多个二进制强极大函数点wise可比较。这些二进制结构定理将Mei和Li、Pipher和Ward的欧几里得类似结果推广到同质类型乘积空间。

Garsia-Rodemich spaces: Local Maximal Functions and Interpolation

We characterize the Garsia-Rodemich spaces associated with a rearrangement invariant space via local maximal operators. Let $Q_{0}$ be a cube in $R^{n}$. We show that there exists $s_{0}\in(0,1),$ such that for all $0<s<s_{0},$ and for all r.i. spaces $X(Q_{0}),$ we have% \[ GaRo_{X}(Q_{0})=\{f\in L^{1}(Q_{0}):\Vert f\Vert_{GaRo_{X}}\simeq\Vert M_{0,s,Q_{0}}^{\#}f\Vert_{X}<\infty\}, \] where $M_{0,s,Q_{0}}^{\#}$ is the Strömberg-Jawerth-Torchinsky local maximal operator. Combined with a formula for the $K-$functional of the pair $(L^{1},BMO)$ obtained by Jawerth-Torchinsky, our result shows th…

紧致域上BMO函数的尖锐体积型界及其应用

Sharp Volume-Based Bounds for BMO Functions on Compact Domains and Applications

本文为紧支撑BMO函数建立以支撑集Lebesgue测度替代直径的尖锐局部$L^r$界，并应用于几何坍缩、薛定谔算子谱界及奇异积分解耦。

Haar 系统 Hardy 空间的独立和中的分解

Factorization in independent sums of Haar system Hardy spaces

本文推广 Bourgain-Rosenthal-Schechtman 空间，定义 Haar 系统 Hardy 空间的独立和 $Y_ω$，并证明其恒等算子可通过大对角算子或其补算子分解。

An operator-valued $T1$ theory for symmetric CZOs

We provide a natural BMO-criterion for the $L_2$-boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the $L_2$-boundedness of the commutators $[R_j,b]$ whenever $b$ belongs to the Bourgain's vector-valued BMO space, where $R_j$ is the $j$-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.

分解问题的维数依赖性：Haar系统Hardy空间

Dimension dependence of factorization problems: Haar system Hardy spaces

本文研究Haar系统Hardy空间上恒等算子的分解问题，证明当维数满足拟多项式或多项式依赖时，任意有界算子或其补算子可分解为两个有界算子的乘积，并给出对角元条件下的改进估计。