L^p - L^q Estimates for parabolic systems in non-divergence form with VMO coefficients

Consider a parabolic N × N-system of order m on ℝⁿ with top-order coefficients a_α ∈ VMO∩L ^∞, Let 1 <p, q <∞ and let ω be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution u satisfying ||u′||_{Lq(J;Lωp(ℝn)N)} + ||Au||_{Lq(J;Lωp,(ℝn)N)} ≤ C||f|| _{Lq(J;Lωp(ℝn)N)}, where Au = Σ _|α|≤ma_αD^αu and J = [0, ∞ ). In particular, choosing ω = 1, the realization of A in L ^p(ℝⁿ)^N has maximal L^pL ^q regularity.