This article gives a necessary and sufficient condition for the invariant Hilbert scheme studied in [10] to be the minimal resolution of a 3-dimensional affine normal quasihomogeneous $\mathit{SL}(2)$-variety.

End-point maximal $L^{1}$-regularity for the parabolic initial-boundary value problem is considered in the half-space. For the inhomogeneous boundary data of both the Dirichlet and the Neumann type, maximal $L^{1}$-regularity for the initial-boundary value problem of parabolic equation is established in time end-point case upon the Besov space as well as the optimal trace estimates. We derive the almost orthogonal properties between the boundary potentials of the Dirichlet and the Neumann boundary data and the Littlewood-Paley dyadic decomposition of unity.