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 ℝ n 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 ′ ∥ L q ( J ; L ω p( ℝ n ) N ) + ∥ A u ∥ L q ( J ; L ω p( ℝ n ) N ) ⩽ C ∥ f ∥ L q ( J ; L ω p( ℝ n ) N ) ,where Au = ∑ |α|⩽ m a α D α u and J = [0,∞). In particular, choosing ω = 1, the realization of A in L p (ℝ n ) N has maximal L p – L q regularity.