This important paper is devoted to the solution of a Dirichlet problem for the Laplace operator in a plane sector. Grisvard’s method and some results in operator theory concerning the closedness of the sum of two operators and the decomposition of a Banach space according to a decomposition of the spectrum of an operator are employed. Main result: The authors present a proof for the existence and uniqueness of solutions in suitably weighted \(W^{2,p}\)-spaces. Moreover a representation formula for the solution is given.