与正交多项式相关的几个问题及其研究

We will investigate function spaces associated with orthogonal polynomials and their applications in proving the boundedness of singular integral operators and well-posedness of wave equations. Including: The characterizations of Hardy spaces associated with orthogonal polynomials by area integral and g-functions; Define BMO space associated with orthogonal polynomials and prove that it is the dual space of Hardy space; The boundedness of Calderon-Zygmund operators associated with orthogonal polynomials on the Hardy spaces and BMO spaces. T(b) theorem for orthogonal polynomials and its application in the proofs of the boundedness of singular integral operators associated with orthogonal polynomials. Definition of Hardy-Sobolev spaces associated with orthogonal polynomials and the various properties of Hardy-Sobolev spaces in the spirit of Sobolev spaces associated with orthogonal polynomials, as applications, we will prove the endpoint versions of the div-curl lemma associated with orthogonal polynomials and give endpoint estimates at p = 1 for the square root of the differential operators associated with orthogonal polynomials. Define Besov spaces associated with orthogonal polynomials and study the well-posedness of wave equations on it. This research projection belongs to central problem of harmonic analysis, it also has much influence on other fields. It is very important both for theory and applications.

本项目主要研究与正交多项式相关的函数空间及其在证明奇异积分算子的有界性和波方程的适定性上的应用，具体包括：与正交多项式相关的Hardy空间的面积积分和g-函数刻画；与正交多项式相关的BMO空间的建立及其与Hardy空间的对偶关系；与正交多项式相关的Calderon-Zygmund算子在相应的Hardy空间和BMO空间上的有界性。与正交多项式相关的T(b)定理及其在证明与正交多项式相关的奇异积分算子的有界性方面的应用。定义与正交多项式相关的Hardy-Sobolev空间并借助与正交多项式相关的Sobolev空间刻画它的性质，作为应用，我们将证明与正交多项式相关的散度-旋度引理的端点情况以及相应微分算子的平方根问题。建立与正交多项式相关的Besov空间并研究相应的波方程在上面的适定性。本研究课题属于调和分析的核心问题，对其它学科分支也具有深远影响，既具有重要的理论意义又具有比较广泛的应用前景。

本项目有四个主要的研究目标：(1) 与正交多项式相关的 Calderon-Zygmund 算子在 Hardy 空间和 BMO 空间上的有界性问题，具体包括 Laguerre 多项式，Hermite 多项式，Bessel 多项式以及特殊 Hermite 多项式等。(2) 利用与正交多项式相关的BMO空间研究相应的T(b)定理并证明与正交多项式相关的奇异积分算子的有界性问题。(3) 定义与正交多项式相关的 Hardy-Sobolev 空间并研究相应的散度-旋度引理及奇异积分算子在上面的有界性问题。(4) 定义与正交多项式相关的Sobolev 空间和Besov 空间并研究与正交多项式相关的微分方程在上面的适定性问题。本项目达到了预期目标。取得的主要成果有：对与特殊Hermite多项式相关的Hardy空间进行了研究，得到了Poisson极大函数刻画，面积积分及g-函数刻画；研究了与退化薛定谔算子相关的Hardy空间并得到了它的面积积分刻画；利用T(b)定理证明了Riesz变换的有界性问题；研究了与Hermite展式以及特殊Hermite展式相关的Hardy-Sobolev空间并证明了散度-旋度引理；证明了关于特殊Hermite展式的Harnack不等式并得到了方程解的Holder估计。