Research on the multi-linear harmonic analysis

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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.