Quasiconformal mappings and chord-arc curves
Quasiconformal mappings and chord-arc curves
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DOI：
10.1090/s0002-9947-1988-0927689-1
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发表时间：
1988
影响因子：
1.3
通讯作者：
S. Semmes
中科院分区：
文献类型：
--
作者：
S. Semmes
Given a quasiconformal mapping p on the plane, what conditions on its dilatation , guarantee that p(R) is rectifiable and PIR is locally abso lutely continuous? We show in this paper that if , satisfies certain quadratic Carleson measure conditions, with small norm, then p(R) is a chord-arc curve with small constant, and p(x) = p(0) + g0 ea(t) dt for x E R, with a E BMO having small norm. Conversely, given any such map from R C, we show that it has an extension to C with the right kind of dilatation. Similar results hold with R replaced by a chord-arc curve. Examples are given that show that it would be hard to improve these results. Applications are given to conformal welding and the theorem of Coifman and Meyer on the real analyticity of the Riemann mapping on the manifold of chord-arc curves. Let p be a quasiconformal map of the plane onto itself. Thus p is a homeomorphism with locally integrable distributional derivatives, and ps = ,upz, where ,u E L°°(C), Il,ul100 < 1. Here we use the notations fz af = AZf = 2 (@Z + id ) f' fZ = 2 (aa ida ) f. This function ,u is called the complex dilatation of p. The mapping theorem for q.c. maps states that for each ,u E L°°(C), Il,ul100 < 1, there is a q.c. map p on C with dilatation ,u, and p is unique up to normalization. A basic problem is to understand how geometric properties of p are reflected in ,u. For example, one would like to have natural conditions on ,u which imply that p(R) is rectifiable and p IR is absolutely continuous. This question arises naturally when considering problems in conformal mappings and conformal welding. In this paper we obtain such estimates for the mapping theorem, and we also give some applications. Our results involve BMO, Aoo weights, chord-arc curves, and Carleson measures, and so we first review the appropriate definitions. A locally integrable function f on R lies in BMO if llfil* = Sup lil | If(x)-fil dx Received by the editors June 1, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 30G60; Secondary 42B20, 30E20.
