Maximal L¹ -regularity for parabolic initial-boundary value problems with inhomogeneous data

End-point maximal L¹-regularity for parabolic initial-boundary value problems is considered. For the inhomogeneous Dirichlet and Neumann data, maximal L¹-regularity for initial-boundary value problems is established in time end-point case upon the homogeneous Besov space B˙p,1s(R+n) with 1 < p< ∞ and - 1 + 1 / p< s≤ 0 as well as optimal trace estimates. The main estimates obtained here are sharp in the sense of trace estimates and it is not available by known theory on the class of UMD Banach spaces. We utilize a method of harmonic analysis, in particular, the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity in the Besov and the Lizorkin–Triebel spaces.