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On boundary value problems for some conformally invariant differential operators

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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, respectively, 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 Lp-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.

Original languageEnglish
JournalCommunications in Partial Differential Equations
Pages (from-to)609-643
Number of pages35
Publication statusPublished - 2016

    Research areas

  • Boundary value problem, Complementary series, Plancherel formula, Poisson transform, Rank one groups, Symmetry breaking operators, Unitary representation

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