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Plemelj-Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators
Univ Manitoba, Dept Math, Machray Hall, Winnipeg, MB R3T 2N2, Canada.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
(English)In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807Article in journal (Refereed) Epub ahead of print
Abstract [en]

Let R be a compact Riemann surface and Gamma be a Jordan curve separating R into connected components Sigma(1) and Sigma(2). We consider Calderon-Zygmund type operators T (Sigma(1), Sigma(k)) taking the space of L2 anti-holomorphic one-forms on Sigma(1) to the space of L-2 holomorphic one-forms on Sigma(k) for k = 1, 2, which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Gamma, to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L2 anti-holomorphic one-forms on R with respect to the inner product on Gamma. We show that the restriction of the Schiffer operator T (Sigma(1), Sigma(2)) to V is an isomorphism onto the set of exact holomorphic one-forms on Sigma(2). Using the relation between this Schiffer operator and a Cauchy-type integral involving Green's function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

Place, publisher, year, edition, pages
SPRINGER HEIDELBERG.
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:uu:diva-397604DOI: 10.1007/s00208-019-01922-4ISI: 000493751200001OAI: oai:DiVA.org:uu-397604DiVA, id: diva2:1372167
Available from: 2019-11-22 Created: 2019-11-22 Last updated: 2019-11-22

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