# zbMATH — the first resource for mathematics

Harmonic measure, $$L^ 2$$ estimates and the Schwarzian derivative. (English) Zbl 0801.30024
Let $$\Phi$$ be a conformal map of the unit disk $$\mathbb{D}$$ onto the inner domain $$\Omega$$ of the Jordan curve $$\Gamma$$; many results also hold in the more general context that $$\Omega$$ is any simply connected domain. This paper can perhaps be considered as a sequel to the authors’ important paper on harmonic measure and arclength [Ann. Math., II. Ser. 132, 511-547 (1992; Zbl 0726.30019)] and contains many further interesting results.
For $$z\in \Gamma$$, $$t>0$$, the authors first study the geometric quantity $\beta(z,t)= \inf_ L t^{-1}\sup\{\text{dist}(w,L): z\in \Gamma\cap D(w,4t)\},{(*)}$ where the infimum is taken over all lines $$L$$ passing through the smaller disk $$D(z,t)$$. They consider the condition $\int^ 1_ 0 t^{-1} \beta(z,t)^ 2 dt< \infty.{(**)}$ If $$K\subset \Gamma$$ is compact and if $$(**)$$ holds uniformly for all $$z\in K$$ then $$K$$ lies on a rectifiable curve. Furthermore, except for a set of linear measure zero, $$\Gamma$$ has a tangent in $$z$$ if and only if $$(**)$$ holds. Next they show that $\iint_ D |\Phi'(z)| | S_ \Phi(z)|^ 2(1- | z|^ 2)^ 3 dx dy< \infty$ implies that $$\Phi'\in H^ p$$ for every $$p< 1/2$$. Here $$S_ \Phi$$ is the Schwarzian derivative that plays an important role in the proofs. Astala and Zinsmeister had characterized when $$\log \Phi'\in \text{BMO}$$. The present authors now give geometric characterizations. One of them (as the reviewer understands it) is: There are constants $$\delta> 0$$ and $$C$$ such that, for every $$z\in \Omega$$, there exists a domain $$G\subset \Omega$$ which is chord-arc (Lavrentiev) with constant $$C$$ such $D(z,\delta d)\subset G,\quad\text{length }\partial G\leq Cd,\quad\text{length }\partial G\cap \partial\Omega\geq \delta d,$ where $$d= \text{dist}(z,\partial\Omega)$$. Their results imply that the condition $$\log \Phi'\in \text{BMO}$$ is invariant under bi-Lipschitz homeomorphisms of the plane and that the condition holds if and only if the corresponding condition holds for the exterior conformal map (in the case of a quasidisk).
There are many details that are difficult to follow. E.g. the definition of $$\beta$$ on p. 86 is different from $$(*)$$ and does not make sense. Saying that “By adjusting the constants in the definitions one gets” some relation leaves doubt in the reader’s mind about that relation. It is stated that Lemma 3.3 was proven as Lemma 3.1 in the authors’ paper quoted above. But the status of that lemma is not quite clear, see M.R. 92c:30026 (loc. cit.). That reviewer writes “Bishop informed the reviewer that Lemma 3.1 can be salvaged by…”.

##### MSC:
 30C85 Capacity and harmonic measure in the complex plane
Full Text:
##### References:
 [1] L. V. Ahlfors,Conformal Invariants, McGraw-Hill, New York, 1973. [2] L. V. Ahlfors,Complex Analysis, McGraw-Hill, New York, 1979. · Zbl 0395.30001 [3] L. V. Ahlfors,Lectures on Quasiconformal Mappings, Wadsworth and Brooks/Cole, Monterey, California, 1987. · Zbl 0605.30002 [4] K. Astala and M. Zinsmeister,Teichmüller spaces and BMOA, Math. Ann.289 (1991), 613–625. · Zbl 0896.30028 · doi:10.1007/BF01446592 [5] A. Baernstein II,Ahlfors and conformal invariants, Ann. Acad. Sci. Fenn13 (1989), 289–312. · Zbl 0682.30026 [6] A. Beurling,Études sur un Problem de Majoration, thesis, Upsala, 1933. [7] C. J. Bishop,Harmonic measures supported on curves, thesis, University of Chicago, 1987. [8] C. J. Bishop,Some conjectures concerning harmonic measure, inPartial Differential Equations with Minimal Smoothness and Applications (B. Dahlberg, E. Fabes, R. Fefferman, D. Jerison, C. Kenig and J. Pipher, eds.), Vol. 42 of IMA Volumes in Mathematics and its Applications, Springer-Verlag, Berlin, 1991. [9] C. J. Bishop, L. Carleson, J. B. Garnett and P. W. Jones,Harmonic measures supported on curves, Pacific. J. Math.138 (1989), 233–236. [10] C. J. Bishop and P. W. Jones,Harmonic measure and arclength, Ann. Math.132 (1990), 511–547. · Zbl 0726.30019 · doi:10.2307/1971428 [11] L. Carleson,Selected Problems on Exceptional Sets, Van Nostrand, Princeton, New Jersey, 1967. · Zbl 0189.10903 [12] L. Carleson,On mappings, conformal at the boundary, J. Analyse Math.19 (1967), 1–13. · Zbl 0186.13701 · doi:10.1007/BF02788706 [13] L. Carleson,On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn. Ser. A.I Math.10 (1985), 113–123. · Zbl 0593.31004 [14] R. Coifman, P. Jones and S. Semmes,Two elementary proofs of the L 2 boundedness of Cauchy integrals on Lipschitz graphs, J. Amer. Math. Soc.2 (1989), 553–564. · Zbl 0713.42010 [15] R. Coifman and Y. Meyer,Lavrentiev’s curves and conformal mappings, Report No. 5, Institute Mittag-Leffer, 1983. [16] R. Coifman and R. Rochberg,Representation theorems for holomorphic and harmonic functions in L p , Asterisque77 (1980), 11–66. · Zbl 0472.46040 [17] B. Dahlberg,On the absolute continuity of elliptic measures, Amer. J. Math.108 (1986), 1119–1138. · Zbl 0644.35032 · doi:10.2307/2374598 [18] G. David,Opérateurs intégraux singuliers sur certains courbes du plan complex, Ann. Sci. École Norm. Sup.17 (1984), 227–239. [19] P. L. Duren,Univalent Functions, Springer-Verlag, New York, 1983. [20] P. L. Duren, H. S. Shapiro and A. L. Shields,Singular measures and domains not of Smirnov type, Duke Math. J.33 (1966), 247–254. · Zbl 0174.37501 · doi:10.1215/S0012-7094-66-03328-X [21] R. Fefferman,A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc.2 (1989), 127–135. · Zbl 0694.35050 · doi:10.1090/S0894-0347-1989-0955604-8 [22] R. Fefferman, C. Kenig and J. Pipher,The theory of weights and the Dirichlet problem for elliptic equation, to appear, J. Amer. Math. Soc. · Zbl 0770.35014 [23] J.-L. Fernandez, J. Heinonen and O. Martio,Quasilines and conformal mappings, J. Analyse Math.52 (1989), 117–132. · Zbl 0677.30012 · doi:10.1007/BF02820475 [24] F. P. Gardiner,Teichmüller Theory and Quadratic Differentials, Wiley, New York, 1987. · Zbl 0629.30002 [25] J. B. Garnett,Bounded Analytic Functions, Academic Press, New York, 1981. · Zbl 0469.30024 [26] J. A. Jenkins,On Ahlfors’ spiral generalization of the Denjoy conjecture, Indiana Univ. Math. J.36 (1987), 41–44. · Zbl 0578.30021 · doi:10.1512/iumj.1987.36.36002 [27] J. A. Jenkins and K. Oikawa,Conformality and semiconformality at the boundary, J. Reine. Angew. Math.291 (1977), 92–117. · Zbl 0339.30024 [28] D. S. Jerison and C. E. Kenig,Hardy spaces, A and singular integrals on chord-arc domains, Math. Scand.50 (1982), 221–248. · Zbl 0509.30025 [29] P. W. Jones,Lipschitz and bi-Lipschitz functions, Rev. Mat. Iberoamericana4 (1988), 115–121. · Zbl 0782.26007 [30] P. W. Jones,Square functions, Cauchy integrals, analytic capacity, and harmonic measure, inHarmonic Analysis and Partial Differential Equations, Lecture Notes in Math.1384, Springer-Verlag, New York, 1989, pp. 24–68. [31] P. W. Jones,Rectifiable sets and the traveling salesman problem, Invent. Math.102 (1990), 1–15. · Zbl 0731.30018 · doi:10.1007/BF01233418 [32] J. P. Kahane,Trois notes sur les ensembles parfait linéaires, Enseigement Math.15 (1969), 185–192. · Zbl 0175.33902 [33] C. Kenig,Weighted H p spaces on Lipschitz domains, Amer. J. Math.102 (1980), 129–163. · Zbl 0434.42024 · doi:10.2307/2374173 [34] N. Kerzman and E. Stein,The Cauchy integral, the Szegö kernel and the Riemann mapping function, Math. Ann.236 (1978), 85–93. · Zbl 0419.30012 · doi:10.1007/BF01420257 [35] M. Lavrentiev,Boundary problems in the theory of univalent functions, Math. Sb.43 (1936), 815–846; Amer. Math. Soc. Transl., Series 232 (1963), 1–35. [36] N. G. Makarov,On distortion of boundary sets under conformal mappings, Proc. London Math. Soc.51 (1985), 369–384. · Zbl 0573.30029 · doi:10.1112/plms/s3-51.2.369 [37] N. G. Makarov,Metric properties of harmonic measure, inProceedings of the International Congress of Mathematicians, Berkeley 1986, Amer. Math. Soc., 1987, pp. 766–776. [38] J. E. McMillan,Boundary behavior of a conformal mapping, Acta. Math.123 (1969), 43–67. · Zbl 0222.30006 · doi:10.1007/BF02392384 [39] K. Okikiolu,Characterization of subsets of rectifiable curves in R n J. London Math. Soc.46 (1992), 336–348. · Zbl 0758.57020 · doi:10.1112/jlms/s2-46.2.336 [40] G. Piranian,Two monotonic, singular, uniformly almost smooth functions, Duke Math. J.33 (1966), 255–262. · Zbl 0143.07405 · doi:10.1215/S0012-7094-66-03329-1 [41] Ch. Pommerenke,Univalent Functions, Vanderhoeck and Ruprecht, Göttingen, 1975. [42] Ch. Pommerenke,On the Green’s function of Fuchsian groups, Ann Acad. Sci. Fen.2 (1976), 409–427. · Zbl 0363.30029 [43] Ch. Pommerenke,On conformal mapping and linear measure, J. Analyse Math.49 (1986), 231–238. · Zbl 0604.30029 · doi:10.1007/BF02796587 [44] F. and M. Riesz,Über die randwerte einer analytischen funltion, inCompte Rendus du Quatrième Congres, des Mathématiciens Scandinaves, Stockholm 1916, Almqvists and Wilksels, Uppsala, 1920. · JFM 47.0295.03 [45] B. Rodin and S. Warschawski,Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn., Ser. AI Math.2 (1976), 467–500. · Zbl 0348.30007 [46] S. Rohde,On conformal welding and quasicircles, Michigan Math. J.38 (1991), 111–116. · Zbl 0726.30010 · doi:10.1307/mmj/1029004266 [47] H. S. Shapiro,Monotonic singular functions of high smoothness, Michigan Math. J.15 (1968), 265–275. · Zbl 0165.06904 · doi:10.1307/mmj/1029000029 [48] E. M. Stein,Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1971. [49] E. M. Stein and A. Zygmund,On differentiability of functions, Studia Math.23 (1964), 247–283. · Zbl 0122.30203 [50] M. Zinsmeister,Représentation conforme et courbes presque lipschitziennes, Ann. Inst. Fourier34 (1984), 29–44. · Zbl 0522.30007 [51] M. Zinsmeister,Domaines Réguliers du plan, Ann. Inst. Fourier35 (1985), 49–55. [52] M. Zinsmeister,Les domains de Carleson, Michigan Math. J.36, 213–220. · Zbl 0692.30030 [53] A. Zygmund,Trigonometric Series, Cambridge University Press, Cambridge, 1959. · Zbl 0085.05601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.