Analysis on Regular Corner Spaces

Abstract

We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel’s calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11–51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective $$2\times 2$$ operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11–51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.