Subelliptic Estimates For Fokker-planck Operator With Variable Coefficients

In this paper we study the Fokker-Planck operator with varialble coefficients in L2 space,for(?),we mainly consider the case that aij is only related to the velocity variable v,which is independent of the time variable t and the spatial variable x,so we can study the properties of P more conveniently by using commutators.the subelliptic estimate of the operator P in the variables x are given under the condition that the coefficient matrix(aij(v))n×n is positive definite and bounded.This paper starts from the following four steps:(1)Firstly,considering the operator about the velocity regularity in L2 space,(?)(2)Then,by applying the methods of commutators and Fourier transform,we get the estimates for(?),(?)(3)Thirdly,constructing the estimates of ?x1/3f,(?)(4)Finally,based on the results of(1)(2)(3),the estimates of ?x2/3f is obtained,which is the subelliptic estimates of the operator P,(?)...