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By Julian Lawrynowicz

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Additional info for Quasiconformal Mappings in the Plane:: Parametrical Methods

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G. ke and Sommer circular onto of multiply on homeomorphic n replaced by in particular, mapping domains of domains bounded by D. of Section 9, ~ canonical Q-qc homeomorphism ~. Consequently, of closed domain D onto this homeomorphism ~. D" of connec- D ~ with the complex diis a desired solution of ( 4 . 2 ) . We still have to prove that the described is determined uniquely apart from conformal mappings selfo Let us suppose fiz) that there exist two solutions which fulfil the conditions of our theorem.

Onto itself Q-quasiconformally with f(O) =0, f(1) :I [and f(~) =~]. e. fractional linear transformations). We are going to formulate the theorem o_~nexistence and uniqueness at first for f ~ S Q . THEOREM 2. Suppose that ~ is a measurable function defined in E and such that II~II~

Is a measurable function defined in II~II~<1. 2). Pro of. 2) with b replaced by b* de- fined as in the proof of Corollary 2. 2). 11) holds, and this can be done exactly in the same way as in the proof of Corollary 2. 2) is replaced with = a(z)w + a*(z)~ + ~(z), are measurable functions with II~II~+ II~*II~<1, while a, a*, and b are locally L p for some p > 2 (for details see Bojarski [2]; for further extensions cf. Bojarski and lwaniec [I]). 10. Extension to multiply connected domains For proving the theorem on existence and uniqueness of qc mappings with a preassumed measurable complex dilatation in the case of multiply connected domains we essentially follow &awrynowicz [I].

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