W_*~(1,p)Regularity for Parabolic Equations Structured on Hrmander's Vector Fields
Zhu Maochu · Chinese Annals of Mathematics · 2014
Let X_1,…X_q(q n) be a family of real smooth vector fields satisfying Hrmander's condition in a bounded domain ΩR~n.We are concerned with the following divergence parabolic equation:u_t+X_i~*(a_(ij)(x,t)X_ju) = X_i~*f_i,where X* is the formal adjoint of X_i,the coefficients a_(ij)(x,t)(i,j = 1,2,…,q) are real valued bounded measurable functions defined in Ω_T = Ω×(O,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.