Higher-order derivative estimates for the heat equation on a smooth domain

We consider the linear heat equation on a bounded domain and on an exterior domain. We study estimates of any order derivatives of the solution locally in time in the Lebesgue spaces. We give a proof of the estimates in the end-point cases $p=1,\mathrm{\infty }$. We also obtain derivative estimates for the equation with the fractional Dirichlet Laplacian.