The Cauchy and the Szegö kernels on multiply connected regions
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LetD be a bounded plane region whose boundary ∂D is Dini-smooth. An operatorB:L2(∂D)→L2(∂D) that has been considered by Kerzman and Stein is investigated. This operator is a compact self-adjoint integral operator whose kernel β (z, ζ) is bounded on ∂Dx∂D and has a geometric description involving chords inD. With the aid ofB the Szegö kernel can be expressed in terms of the Cauchy kernel. Here, the operatorB is recovered from the classical theory of kernel functions resulting in extension of the work of Kerzman and Stein. In particular, the eigenvalue problem associated with the operatorB is studied and its complete equivalence with a very classical problem due to Bergman, Schiffer and Singh is established.
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