Jones, Gavin L. Conformal welding of Jordan curves using weighted Dirichlet spaces. (English) Zbl 0970.30009 Ann. Acad. Sci. Fenn., Math. 25, No. 2, 413-415 (2000). 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). Reviewer: S.V.Kislyakov (St.Peterburg) Cited in 1 Document MSC: 30C55 General theory of univalent and multivalent functions of one complex variable 30C70 Extremal problems for conformal and quasiconformal mappings, variational methods Keywords:normal family; comformal map Citations:Zbl 0193.03703 PDFBibTeX XMLCite \textit{G. L. Jones}, Ann. Acad. Sci. Fenn., Math. 25, No. 2, 413--415 (2000; Zbl 0970.30009) Full Text: EuDML EMIS