Almost orthogonality in harmonic analysis and its application

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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.