Maximal regularity for the Cauchy problem of the heat equation in BMO

Abstract

We consider maximal regularity for the Cauchy problem of the heat equation in a class of bounded mean oscillations (B M O). Maximal regularity for non-reflexive Banach spaces is not obtained by the established abstract theory. Based on the symmetric characterization of B M O-expression, we obtain maximal regularity for the heat equation in B M O and its sharp trace estimate. Our result shows that the homogeneous initial estimate obtained by Stein [50] and Koch–Tataru [32] can be strengthened up to the inhomogeneous estimate for the external forces and the obtained estimates can be applicable to quasilinear problems. Our method is based on integration by parts and can also be applicable to other type of parabolic problems.