  In this paper, we consider the Laplace operator on the half-space with
Dirichlet and Neumann boundary conditions. We prove that this operator admits a
bounded $H^\infty$-calculus on Sobolev spaces with power weights measuring the
distance to the boundary. These weights do not necessarily belong to the class
of Muckenhoupt $A_p$ weights. We additionally study the corresponding Dirichlet
and Neumann heat semigroup. It is shown that these semigroups, in contrast to
the $L^p$-case, have polynomial growth. Moreover, maximal regularity results
for the heat equation are derived on inhomogeneous and homogeneous weighted
Sobolev spaces.
