On boundary value problems for some conformally invariant differential operators

TL;DR

This paper investigates boundary value problems for conformally invariant differential operators on Euclidean space and the Heisenberg group, establishing self-adjointness, unique solutions, explicit Poisson transforms, and their properties in Sobolev and L^p spaces.

Contribution

It introduces new boundary value problem solutions for conformally invariant operators and explicitly constructs their Poisson transforms using representation theory.

Findings

Operators are self-adjoint in Sobolev spaces

Unique solutions to boundary value problems are proven

Explicit Poisson transforms are derived and shown to be isometric

Abstract

We study boundary value problems for some differential operators on Euclidean space and the Heisenberg group which are invariant under the conformal group of a Euclidean subspace resp. Heisenberg subgroup. These operators are shown to be self-adjoint in certain Sobolev type spaces and the related boundary value problems are proven to have unique solutions in these spaces. We further find the corresponding Poisson transforms explicitly in terms of their integral kernels and show that they are isometric between Sobolev spaces and extend to bounded operators between certain $L^p$-spaces. The conformal invariance of the differential operators allows us to apply unitary representation theory of reductive Lie groups, in particular recently developed methods for restriction problems.