Analysis in weighted spaces: functional calculus and parabolic boundary value problems

In this talk, we discuss the development of well-posedness and maximal regularity theory for parabolic boundary value problems in the setting of weighted function spaces. We study prototype differential operators, such as the Laplace operator, on Sobolev spaces with power weights measuring the distance to the boundary of the domain. Using the weighted spaces, we obtain a more flexible well-posedness theory that avoids certain restrictive conditions required in the unweighted setting. We discuss the fundamental properties of weighted function spaces, as well as maximal regularity and the boundedness of the functional calculus for the Laplace operator on these spaces. Finally, we demonstrate applications to (stochastic) partial differential equations and, in particular, parabolic boundary value problems with rough data. This talk is based on joint projects with Robert Denk, Nick Lindemulder, Emiel Lorist and Mark Veraar.