Abstract
In 1981, Hayman and Wu proved that for any simply connected domain Ω and any Riemann mappingF: Ω →D,F′ ∈ L1 (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′ ∈ L1+∈(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′ ∈ L1+∈(Γ ∩ Ω). This result is the synthesis of the extension of Fernández, Heinonen and Martio, and the result of Bishop and Jones showing thatF′ ∈ L1(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.
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References
L. Ahlfors,Complex Analysis (3rd ed.), McGraw-Hill, New York, 1979.
A. Baernstein II,A counterexample concerning integrability of derivatives of conformal mappings, J. Analyse Math.53 (1989), 253–272.
A. Beardon,Geometry of Discrete Groups, Springer-Verlag, New York, 1983.
C. Bishop and P. W. Jones,Harmonic measure and arclength, Ann. of Math.132 (1990), 511–547.
C. Bishop and P. W. Jones,Harmonic measure, L2 estimates and the Schwarzian derivative, J. Analyse Math. (to appear).
L. Carleson,Estimates of Harmonic Measures, Ann. Acad. Sci. Fenn. Ser. A I. Math.7 (1982), 25–32.
R. R. Coifman and C. Fefferman,Weighted norm inequalities for maximal functions and singular integrals, Studia Math.51 (1974), 241–250.
J.-L. Fernández and D.H. Hamilton,Lengths of curves under conformal mapping, Comm. Math. Helv.62 (1987), 122–134.
J.-L. Fernández, J. Heinonen, and O. Martio,Quasilines and conformal mappings, J. Analyse Math.52 (1989), 117–132.
J.-L. Fernández and M. Zinsmeister,Ensembles de niveau des representations conformes, C. R. Acad. Sci. Paris305 (1987), 449–452.
J.B. Garnett,Bounded Analytic Functions, Academic Press, New York, 1983.
J.B. Garnett,Applications of Harmonic Measure, John Wiley & Sons, New York, 1986.
F. W. Gehring,The L P -integrability of the partial derivatives of a quasiconformal mapping, Acta Math.130 (1973), 265–277.
W. K. Hayman and G. Wu,Level sets of univalent functions, Comm. Math. Helv.56 (1981), 366–403.
P. W. Jones,Rectifiable sets and the traveling Salesman problem, Invent. Math.102 (1990), 1–15.
D. S. Jerison and C. E. Kenig,Hardy spaces, A ∞ and singular integrals on chord-arc domains, Math. Scand.50 (1982), 221–248.
Ch. Pommerenke,Univalent Functions, Vanderhoeck and Ruprecht, Göttingen, 1975.
G. Piranian and A. Weitsman,Level sets of infinite length, Comm. Math. Helv.53 (1978), 161–164.
K. Øyma,Harmonic measure and conformal length, Proc. Amer. Math. Soc. (to appear).
K. Øyma,The Hayman-Wu constant, Proc. Amer. Math. Soc. (to appear).
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Walden, B.L. L P-Integrability of derivatives of Riemann mappings on Ahlfors-David regular curves. J. Anal. Math. 63, 231–253 (1994). https://doi.org/10.1007/BF03008425
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DOI: https://doi.org/10.1007/BF03008425