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

摘要

Consider a parabolic N x N-system of order m on R-n with top-order coefficients a(alpha) is an element of VMO boolean AND L-infinity. Let 1 < p: q < infinity and let omega be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution u satisfying parallel to u'parallel to L-q(J;L omega P(Rn)N) + parallel to Au parallel to(Lq(J;L omega P(Rn)N)) <= C parallel to f parallel to(Lq(J;L omega p(Rn)N)), where Au = Sigma(vertical bar alpha vertical bar <= m) a(alpha)D(alpha)u and J = [0, infinity). In particular, choosing omega = 1, the realization of A in L-p(R-n)(N) has maximal L-p - L-q regularity.