The heat equation with rough boundary conditions and holomorphic functional calculus

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H$^{∞}$-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.