Weighted Function Spaces with Applications to Boundary Value Problems

Abstract

This thesis is concerned with the maximal regularity problem for parabolic boundary value problems with inhomogeneous boundary conditions in the setting of weighted function spaces and related function space theoretic problems.
This in particularly includes weighted $L_{q}$-$L_{p}$-maximal regularity but also weighted $L_{q}$-maximal regularity in weighted Triebel-Lizorkin spaces.
The weights we consider are power weights in time and in space, and yield flexibility in the optimal regularity of the initial-boundary data and allow to avoid compatibility conditions at the boundary.
Moreover, the use of scales of weighted Triebel-Lizorkin spaces also provides a quantitative smoothing effect for the solution on the interior of the domain.