Boundary value problems of elliptic and parabolic type with boundary data of negative regularity

We study elliptic and parabolic boundary value problems in spaces of mixed scales with mixed smoothness on the half-space. The aim is to solve boundary value problems with boundary data of negative regularity and to describe the singularities of solutions at the boundary. To this end, we derive mapping properties of Poisson operators in mixed scales with mixed smoothness. We also derive $$\mathcal {R}$$ R -sectoriality results for homogeneous boundary data in the case that the smoothness in normal direction is not too large.