Walden, B.L. J. Anal. Math. (1994) 63: 231. doi:10.1007/BF03008425

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Abstract

In 1981, Hayman and Wu proved that for any simply connected domain Ω and any Riemann mappingF: Ω →D,F′ ∈ L^{1} (L ∩ Ω), whereL is any line in the complex plane. Several years later, Fernández, Heinonen and Martio showed that there is anε > 0 such thatF′ ∈ L^{1+∈}(L ∩ Ω). The question arises as to which curves other than lines satisfy such a statement. A curve Γ is said to be Ahlfors-David regular if there is a constantA such that for any B(x, r) (the disk of radiusr centered atx), l(Γ ∩ B(x, r))≤ Ar. The major result of the paper is the following theorem: Let Γ be an Ahlfors-David regular curve with constantA. Then there exists an∈ > 0, depending only onA, such thatF′ ∈ L^{1+∈}(Γ ∩ Ω). This result is the synthesis of the extension of Fernández, Heinonen and Martio, and the result of Bishop and Jones showing thatF′ ∈ L^{1}(Γ ∩ Ω). The proof of the results uses a stopping-time argument which seeks out places in the curve where small pieces may be added in order to control the portions of the curve where ¦F′ ¦ is large. This is accomplished with an estimate on the vanishing of the harmonic measure of the curve in such places. The paper also includes simpler arguments for the special cases where Γ = ∂Ω and Γ ⊂Ω.