  In this paper we consider the Laplace operator with Dirichlet boundary
conditions on a smooth domain. We prove that it has a bounded
$H^\infty$-calculus on weighted $L^p$-spaces for power weights which fall
outside the classical class of $A_p$-weights. Furthermore, we characterize the
domain of the operator and derive several consequences on elliptic and
parabolic regularity. In particular, we obtain a new maximal regularity result
for the heat equation with rough inhomogeneous boundary data.
