On maximal regularity of typeLp–Lqunder minimal assumptions for elliptic non-divergence operators

Abstract

We prove maximal regularity of type L p -L q for operators in non-divergence form with complex-valued measurable coefficients on R n . For a certain range of q, which depends on dimension and the order of the operators, this is done under the sole assumption that they generate an analytic semigroup in L q . Thus we give, for this class of operators and this range of q, a positive answer to Brézis' question whether generation of an analytic semigroup entails maximal L p -regularity. For other values of q we give several additional assumptions. The proof relies on a result on maximal regularity under the assumption of suitable off-diagonal bounds due to S. Blunck and the author, which we improve here. These off-diagonal bounds are obtained by a modification of Davies' technique suited to cope with operators in non-divergence form, and they also imply new results on the scale of L q -spaces in which a given operator generates an analytic semigroup. We thus obtain a completely new approach to maximal L p -regularity for operators of this kind, in particular for those whose highest order coefficients belong to VMO. Moreover, we obtain extensions of recent results due to Kim and Krylov for operators with measurable coefficients depending on one coordinate.

Maximal regularity and applications to PDEs

“…This result was generalised by P.C. Kunstmann [9] for negative generators of semigroups with integrated gaussian estimates.…”