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Journal of the London Mathematical Society 2006 74(3):717-736; doi:10.1112/S0024610706023192
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© The London Mathematical Society

Lp – Lq Estimates for Parabolic Systems in Non-Divergence Form with VMO Coefficients

Robert Haller-Dintelmann, Horst Heck and Matthias Hieber

Technische Universität Darmstadt Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germany haller{at}mathematik.tu-darmstadt.de
Technische Universität Darmstadt Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germany heck{at}mathematik.tu-darmstadt.de
Technische Universität Darmstadt Fachbereich Mathematik, Schlossgartenstraße 7, D-64289 Darmstadt, Germany hieber{at}mathematik.tu-darmstadt.de

Received 13 September 2005.

Consider a parabolic NxN-system of order m on Rn with top-order coefficients a{alpha} VMO{cap}L{infty}. Let 1 < p, q < {infty} and let {omega} be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution u satisfying

Formula
where Au = {sum}|{alpha}|≤m a{alpha} D{alpha}u and J = [0,{infty}). In particular, choosing {omega} = 1, the realization of A in Lp(Rn)N has maximal Lp – Lq regularity.


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