W_*~(1,p) Regularity for Parabolic Equations Structured on H?rmander’s Vector Fields

摘要

Let X_1, …，X_q (q < n) be a family of real smooth vector fields satisfying H?rmander’s condition in a bounded domain Ω ? R~n. We are concerned with the following divergence parabolic equation: ut+X_i~* (a_(ij) (x, t) X_ju) = X_i~* f_i, where X_i~* is the formal adjoint of X_i, the coefficients aij(x,t) (i,j = 1,2, …,q) are real valued bounded measurable functions defined in Ω_T = Ω x （0，T） ∈ R_~(n+1), satisfying the uniform ellipticity condition. Under the assumption that the coefficients a_(ij)(x, t) belong to the space VMO_(loc) ∩ L~∞, the interior W_*~(1,p) regularity for weak solutions of the equation above is established.