Interior W∗1,p(x,t)-regularity for parabolic divergence equations of Hörmander’s type

Let X1,X2,…,XqX1,X2,…,Xq be a system of real smooth vector fields satisfying Hörmander’s condition in a bounded domain Ω⊂RnΩ⊂Rn. We consider the following parabolic equationsut+Xi∗(aij(x,t)Xju)=Xi∗fiinΩT, where Xi∗ is the formal adjoint of XiXi, the coefficients aij(x,t)aij(x,t) are real valued measurable functions defined in ΩT=Ω×(0,T]ΩT=Ω×(0,T], satisfying the uniform parabolic conditionequation(0.1)μ|ξ|2≤∑i,j=1qaij(x,t)ξiξj≤μ−1|ξ|2, for almost every (x,t)∈Rn×R(x,t)∈Rn×R, every ξ∈Rqξ∈Rq and some constant μμ. We prove the interior W∗1,p(x,t)-regularity for weak solutions to the parabolic equations, under the assumptions that p(x,t)p(x,t) satisfies the strong log-Hölder continuity condition and the coefficients aij(x,t)aij(x,t) belong to the space VMOloc∩L∞VMOloc∩L∞. Our method relies on a Gagliardo–Nirenberg inequality constituted on Hörmander’s vector fields and a certain Vitali covering lemma.