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Conformal welding of Jordan curves using weighted Dirichlet spaces. (English) Zbl 0970.30009

For a Jordan curve \(J\) on the Riemann sphere, consider conformal maps \(f\) and \(g\) of the unit disk onto the inner and the outer domain of \(J\), respectively. Then \(g^{-1}\circ f\) is a homeomorphism of the unit circle \(S^1\) called a welding of \(J\).
Extending the results of O. Lehto [see Proc. 15th Scand. Congr. Oslo, 1968, Lect. Notes Math. 118, 58-73 (1970; Zbl 0193.03703)], the author shows that for a homeomorphism \(h: S^1\to S^1\) to be a welding of some Jordan curve, it suffices that \(h\) be the uniform limit of homeomorphisms \(h_n\) such that the composition operators \(f\mapsto f\circ h_n\) are uniformly bounded from a certain weighted Dirichlet space to the usual Dirichlet space. Some applications are discussed, in particular, to the problem of uniqueness of a welding (up to a Möbius transformation).

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods

Citations:

Zbl 0193.03703
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