  End-point maximal $L^1$-regularity for parabolic initial-boundary value
problems is considered. For the inhomogeneous Dirichlet and Neumann data,
maximal $L^1$-regularity for initial-boundary value problems is established in
time end-point case upon the homogeneous Besov space $\dot B_{p,1}^s({\mathbb
R}^n_+)$ with $1< p< \infty$ and $-1+1/p<s\le 0$ 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.
