L~p-L~q Estimates For A Class Of Pseudo-differential Equations And Its Applications

Since 1920's, Schr(o|¨)dinger equations with abundant theoretic and applied backgrounds have been constantly central subjects of study of mathematics and physics. The study of higher-order Schr(o|¨)dinger equations, which are natural generalization of Schr(o|¨)dinger equations not only can promote the development of mathematic itself, but also deepen the comprehension of Schr(o|¨)dinger equations.This paper is concerned with the L~p-L~q estimates of the solutions for a class of pseudo-differential equations under some degenerate assumptions. As applications, these estimates can be used to show generalized Schr(o|¨)dinger operator with some integrable potential generates a fractionally integrated group in L~p(R~n).In this paper, we mainly use the methods and techniques of harmonic analysis, meanwhile, some tools and means of Functional analysis also be used. All of them include interpolation of operator, Fourier transforms carried on surface, decomposition of the unit and fractional integral semigroup. Especially, we use the methods from the oscillatory integrals which are frequently used in Schr(o|¨)dinger equations.