A characterization of weighted Sobolev spaces and its applications

点击翻译按钮获取中文摘要

1.We proved a weighted version of the Sobolev-Lieb-Thirring inequality. As an application we showed a weighted LP-Sobolev-Lieb-Thirring inequality which is a new result even in the non-weighted case.2.We proved that the wavelet basis is a greedy basis in weighted Triebel-Lizorkin spaces. As an application we determined the approximation spaces of the weighted Triebel-Lizorkin spaces by means of the non-linear approximation by wavelets.3.We proved a wavelet characterization of weighted Herz spaces. Tang and Yang showed a vector valued inequality in weighted Herz space although there are some mistakes in their condition on weights. We gave a correct condition on weights and proved the wavelet characterization.4.We studied the decay estimate of the solution of a non-linear elliptic partial differential equations with a potential. We characterize the class of weights in the decay estimate of the solution by the potential. We proved the global existence of a solution with small amplitude for several dispersive equations such as modified Boussinesq equations, improved Boussinesq equations and semi relativistic Hartree equations.5.We studied the existence of a singular solution and its property of a semilinear degenerate elliptic equation with p-harmonic operator in the principal term. In particular we investigated a linearized degenerate elliptic operator and proved the non-negativeness of the smallest eigenvalue and its relation to the Hardy type inequality.