Almost orthogonality in harmonic analysis and its application

本课题研究了用相空间局域函数分析几种算子的方法。首先，利用Gabor坐标系推广了关于伪微分算子L^2有界性的Calderon-Vaillancourt定理。其次，利用Bellman函数方法证明了离散并矢Carleson不等式。作为应用，我们给出了分数阶极大算子和分数阶积分算子的加权范数不等式的交替证明。第三，我们得到了关于负势薛定谔算子负特征值矩的Lieb-Thirring不等式的推广。我们用了fraier - jawerth的ψ变换。所得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可以应用于物质的稳定性问题和非线性方程吸引子的豪斯多夫维数的估计。第四，研究了从D维单位球B_1(0)到（D+1）维欧几里德空间的调和映射的非线性热方程弱解的正则性结构和奇异集。我们证明了谐波映射的最小值除了不超过(d-3)-Hausdorff维数的闭集之外是光滑的。

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.