谱乘子与函数空间的二进结构研究

It is one of important subjects in harmonic analysis to study restriction theorems associated with differential operators, Bochner-Riesz means and spectral multiplier theory. This project intends to study these topics combining classical harmonic analysis method and the recent development of the singular integral operators with non-smooth kernel. For a class of degenerate elliptic differential operators, we will study the weighted restriction estimates and spectral multipliers associated with these operators. Also we will study sharp L ^ p and H ^ p (Hardy spaces associated with operators) boundedness of Bochner-Riesz means associated with these operators. Meanwhile, the project also intends to study spectral multiplier theorems on product spaces..Random dyadic cubes method solves some significant open problems, such as the two-weight problem for Hilbert transform and the A2 conjecture. The project also intends to carry out the research of dyadic structure of Hardy spaces and BMO spaces under the single parameter and multi-parameter cases. Based on these two parts of research, we will try to study dyadic function spaces associated with operators and A2 conjecture for spectral multipliers ( singular integral operators with non-smooth kernel).

与微分算子相联的限制性定理、Bochner-Riesz平均有界性问题与谱乘子理论是调和分析的前沿研究课题之一。本项目拟结合经典调和分析方法和近年来发展的非光滑核奇异积分算子理论研究这一课题。拟针对一类退化椭圆微分算子，研究与它相联的加权限制性估计和谱乘子定理以及与它相联的Bochner-Riesz平均的最佳L^p估计和H^p(与算子相联的Hardy空间)估计。同时，本项目还拟将单参数的谱乘子定理推广到乘积空间的情形。. 随机二进方体方法的成熟应用解决了诸如Hilbert变换双权问题和A2权最佳常数问题等公开问题。本项目拟开展单参数和多参数情形下Hardy空间与BMO空间的二进结构的研究。以上述两部分研究为基础，我们还将尝试研究与算子相联的二进函数空间和谱乘子（非光滑核）的A2权最佳常数问题。

项目组较为圆满的完成了原有的研究目标，发表SCI论文8篇，其中包括《J. Funct. Anal.》、《Math. Z.》等国际权威期刊。. 谱乘子方面，首先针对一般非负自伴算子，利用与算子相联的限制性估计得到了最佳谱乘子定理和Bochner-Riesz平均算子的端点估计。其次，针对具体的微分算子，如类调和震荡子和Grushin型算子，得到最佳的谱乘子定理和Bochner-Riesz平均的有界性。另外针对乘积空间，得到最佳的Marcinkiewicz型谱乘子定理。同时，我们还研究了双线性粗糙核奇异积分算子的权常数的问题。. 函数空间方面，首先我们研究了与算子相联的乘积型Hardy空间及其各种刻画和插值定理。其次，我们研究了与Schrodinger算子有关的BMO空间的balayages分解刻画。还有，我们研究了经典的CMO空间和双线性奇异积分算子交换子紧性之间的等价刻画。