Research on the multi-linear harmonic analysis

在本研究中，我们利用Frazier和Jawerth的ψ变换研究了加权函数空间的刻画及其对偏微分算子的应用。我们证明了Lieb-Thirring不等式的一个推广，它给出了具有负势的薛定谔算子的负本征值的矩的估计。我们的结果是关于高阶退化椭圆算子的。我们推广了Egorov-Kondrat‘ev的结果，证明了Sobolev-Lieb-Thirring不等式的推广。我们的结果是原始结果的加权版本，它是Gmidaglia-Marion-Temam定理或Edmunds-Ilyin定理的推广。我们的结果有望应用于非线性偏微分算子吸引子的Hausdorff维数的估计。

In this research we studied about the characterization of weighted function spaces by means of the ψ-transform of Frazier and Jawerth and its appilications to partial differential operators. We proved a generalization of the Lieb-Thirring inequality which gives an estimate of the moments of the negative eigenvalues of the Schrodinger operator with negative potentials. Our result is about higher order degenerate elliptic operators. We gave a generalization of Egorov-Kondrat'ev's result.We also proved a generalization of the Sobolev-Lieb-Thirring inequality. Our result is a weighted version of the original one, which is a generalization of Ghidaglia-Marion-Temam's theorem or Edmunds-Ilyin's theorem. Our results are expected to be applied to the estimate of the Hausdorff dimension of the attractor of nonlinear partial differential operators.