Partial differential equations and, in particular, boundary value problems model a wide variety of physical, chemical and biological processes. Often these processes exhibit a certain singular behaviour near the boundary of the underlying domain, for example due to a non-smooth geometry of this boundary or due to the presence of noise. Mathematically these singularities can be captured by working in weighted Lebesgue spaces with weights that are suitable powers of the distance to the boundary. However, the typical weight class for which standard harmonic analysis is available does not suffice for this purpose. In this talk we discuss holomorphic functional calculus for the heat equation subject to the Dirichlet boundary condition in such a weighted setting. This functional calculus can be viewed as an extension of the spectral theory of self-adjoint operators on Hilbert spaces.