Almost orthogonality in harmonic analysis and its application

在这个项目中，我们研究了分析的几个运营商的功能，这是本地化的相空间。首先，我们利用Gabor框架得到了关于伪微分算子L^2有界性的Calderon-Vaillancourt定理的一个推广。其次，利用Bellman函数方法证明了离散并矢Carleson不等式。作为应用，给出了分数次极大算子和分数次积分算子的加权范数不等式的交替证明。第三，得到了关于具有负位势的Schrodinger算子的负特征值矩的Lieb-Thirring不等式的推广。我们使用了Frazier-Jawerth变换。我们的结果适用于高阶退化椭圆型偏微分算子。我们期望我们的结果对物质的稳定性和非线性方程吸引子的Hausdorff维数的估计问题有一定的应用。第四，我们研究了d维单位球B_1（0）到（D+1）维欧氏空间调和映射所对应的非线性热方程弱解的正则性结构和奇异集。证明了调和映射的极小元除了至多（d-3）-Hausdorff维数的闭集外都是光滑的。

In this project we studied the analysis of several operators by means of functions which are localized in phase space. First, we got a generalization of Calderon-Vaillancourt's theorem about the L^2 boundedness of pseudodifferential operators by means of Gabor frames. Second, we proved the descrete dyadic Carleson's inequality by using Bellman function method. As an application we gave alternate proofs of weighted norm inequalities for fractional maximal operators and fractional integral operators, Third, we got a generalization of Lieb-Thirring inequality about the moments of negative eigenvalues of the Schrodinger operator with negative potential. We used Frazier-Jawerth's ψ-transform. Our result is applicable to higher order degenerate elliptic partial differential operators. We expect that our result has an application to the problem of the stability of matter and the estimate of the Hausdorff dimension of the attractor of nonlinear equations. Four, we investigated the structure of regularity and the singular set of the weak solution of the nonlinear heat equaltion associated with a harmonic map from d-dimensional unitball B_1 (0) to (D+1)-dimensional Euclidean space. We showed that the minimizer of the harmonic map is smooth except closed set of at most (d-3)-Hausdorff dimension.