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On the Analyticity of the Schwarz Operator with Respect to a Curve

In: Factorization, Singular Operators and Related Problems

Author

Listed:
  • Luca Preciso

    (University of Padova)

  • Sergei Rogosin

    (Belarusian State University)

Abstract

The classical Schwarz operator T[•, •, •] assigns to each triple (ø, f, w), where ø is a plane closed curve enclosing a simply connected domain D, f is a real-valued function of the boundary of D and w ∈ D, the unique holomorphic function F of D satisfying Re F = f on the boundary of D and Im F(w) = 0. We consider the modified Schwarz operator T *, T * [ø, p, w] ≡ T[ø, p o ø(−1) w]o ø, which assigns to each triple (ø, p, w) the composition of the classical Schwarz operator T valued in (ø, poø (−1), w) with the boundary curve ø. We show that T *. depends real analytically on its variables and compute the first differential of T *. Regularity of another type of modifications of the Schwarz operator T is studied too.

Suggested Citation

  • Luca Preciso & Sergei Rogosin, 2003. "On the Analyticity of the Schwarz Operator with Respect to a Curve," Springer Books, in: Stefan Samko & Amarino Lebre & António F. dos Santos (ed.), Factorization, Singular Operators and Related Problems, pages 237-254, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-0227-0_16
    DOI: 10.1007/978-94-017-0227-0_16
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