Carleman-Sobolev classes and Green’s potentials for weighted Laplacians

Sammanfattning: This thesis is based on two papers: the first one concerns Carleman-Sobolev classes for small exponents and the other solves Poisson's equation for the standard weighted Laplacian in the unit disc.In the first paper we start by noting that for small Lp-exponents, i.e. 0<p<1, the way we usually define Sobolev spaces is very unsatisfactory, which was illustrated by Peetre in 1975. In an attempt to remedy this we introduce completions of a class of smooth functions, which we call Carleman-Sobolev classes since they generalize Sobolev spaces and uses a norm inspired by Carleman classes. If the class is restricted with a growth condition on the supremum norms of the derivatives, we prove that there exists a condition on the weight sequence in the norm which guarantees that the resulting completion can be embedded into C?(R). This condition is even sharp up to some regularity on the weight sequence, in the sense that the norm inequality required for continuity no longer holds. We also show that the growth condition is necessary, in the sense that if we drop it entirely we can naturally embed Lp into this class's completion. Hence in this case we cannot consider the completion as a proper generalization of a Sobolev space.In the second paper we find Green's function for the standard weighted Laplacian and give conditions on the Riesz-mass such that we can use Green's potential to solve Poisson's equation with zero boundary values in the sense of radial L1-means. The weight here comes from the theory of weighted Bergman spaces and from this context it gets the label as the standard weight.