Cauchy transforms of self-similar measures: Starlikeness and univalence
HTML articles powered by AMS MathViewer
- by Xin-Han Dong, Ka-Sing Lau and Hai-Hua Wu PDF
- Trans. Amer. Math. Soc. 369 (2017), 4817-4842 Request permission
Abstract:
For the contractive iterated function system $S_kz=e^{2\pi ik/m}+{\rho (z-e^{2\pi ik/m})}$ with $0<\rho <1, k=0,\cdots , m-1$, we let $K\subset \mathbb {C}$ be the attractor, and let $\mu$ be a self-similar measure defined by $\mu =\frac 1m\sum _{k=0}^{m-1}\mu \circ S_k^{-1}$. We consider the Cauchy transform $F$ of $\mu$. It is known that the image of $F$ at a small neighborhood of the boundary of $K$ has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of $F$ away from $K$; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures.References
- Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
- Vasilis Chousionis, Singular integrals on Sierpinski gaskets, Publ. Mat. 53 (2009), no. 1, 245–256. MR 2474123, DOI 10.5565/PUBLMAT_{5}3109_{1}1
- Vasilis Chousionis, Valentino Magnani, and Jeremy T. Tyson, Removable sets for Lipschitz harmonic functions on Carnot groups, Calc. Var. Partial Differential Equations 53 (2015), no. 3-4, 755–780. MR 3347479, DOI 10.1007/s00526-014-0766-1
- Vasilis Chousionis and Pertti Mattila, Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups, J. Reine Angew. Math. 691 (2014), 29–60. MR 3213547, DOI 10.1515/crelle-2012-0078
- Vasileios Chousionis and Mariusz Urbański, Homogeneous kernerls and self-similar sets, Indiana Univ. Math. J. 64 (2015), no. 2, 411–431. MR 3344433, DOI 10.1512/iumj.2015.64.5491
- Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
- Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy transform, Mathematical Surveys and Monographs, vol. 125, American Mathematical Society, Providence, RI, 2006. MR 2215991, DOI 10.1090/surv/125
- Guy David, Unrectifiable $1$-sets have vanishing analytic capacity, Rev. Mat. Iberoamericana 14 (1998), no. 2, 369–479 (English, with English and French summaries). MR 1654535, DOI 10.4171/RMI/242
- Xinhan Dong, Cauchy transforms of self-similar measures, ProQuest LLC, Ann Arbor, MI, 2002. Thesis (Ph.D.)–The Chinese University of Hong Kong (Hong Kong). MR 2703423
- Xin-Han Dong and Ka-Sing Lau, Cauchy transforms of self-similar measures: the Laurent coefficients, J. Funct. Anal. 202 (2003), no. 1, 67–97. MR 1994765, DOI 10.1016/S0022-1236(02)00069-1
- Xin-Han Dong and Ka-Sing Lau, An integral related to the Cauchy transform on the Sierpinski gasket, Experiment. Math. 13 (2004), no. 4, 415–419. MR 2118265, DOI 10.1080/10586458.2004.10504549
- Xin-Han Dong and Ka-Sing Lau, Cantor boundary behavior of analytic functions, Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2010, pp. 283–294. MR 2743000, DOI 10.1007/978-0-8176-4888-6_{1}8
- X. H. Dong and K. S. Lau, Cauchy Transform on Sierpinski Gasket: fractal behavior at the boundary, submitted.
- Xin-Han Dong, Ka-Sing Lau, and Jing-Cheng Liu, Cantor boundary behavior of analytic functions, Adv. Math. 232 (2013), 543–570. MR 2989993, DOI 10.1016/j.aim.2012.09.021
- Peter L. Duren, Univalent functions, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
- Kenneth Falconer, Fractal geometry, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. Mathematical foundations and applications. MR 2118797, DOI 10.1002/0470013850
- John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. MR 0276456, DOI 10.1090/S0002-9939-1970-0276456-5
- John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006, DOI 10.1007/BFb0060912
- D. J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math. Soc. 192 (1974), 285–292. MR 338338, DOI 10.1090/S0002-9947-1974-0338338-8
- W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. MR 1310776, DOI 10.1017/CBO9780511526268
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
- Ka-Sing Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), no. 2, 335–358. MR 1239075, DOI 10.1006/jfan.1993.1116
- Ka-Sing Lau and Sze-Man Ngai, Multifractal measures and a weak separation condition, Adv. Math. 141 (1999), no. 1, 45–96. MR 1667146, DOI 10.1006/aima.1998.1773
- Jing-Cheng Liu, Xin-Han Dong, and Shi-Mao Peng, A note on Cantor boundary behavior, J. Math. Anal. Appl. 408 (2013), no. 2, 795–801. MR 3085074, DOI 10.1016/j.jmaa.2013.06.059
- John-Peter Lund, Robert S. Strichartz, and Jade P. Vinson, Cauchy transforms of self-similar measures, Experiment. Math. 7 (1998), no. 3, 177–190. MR 1676691, DOI 10.1080/10586458.1998.10504368
- Thomas H. MacGregor, The radius of convexity for starlike functions of order ${1\over 2}$, Proc. Amer. Math. Soc. 14 (1963), 71–76. MR 150282, DOI 10.1090/S0002-9939-1963-0150282-6
- Joan Mateu, Xavier Tolsa, and Joan Verdera, The planar Cantor sets of zero analytic capacity and the local $T(b)$-theorem, J. Amer. Math. Soc. 16 (2003), no. 1, 19–28. MR 1937197, DOI 10.1090/S0894-0347-02-00401-0
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Pertti Mattila, Smooth maps, null-sets for integralgeometric measure and analytic capacity, Ann. of Math. (2) 123 (1986), no. 2, 303–309. MR 835764, DOI 10.2307/1971273
- Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-$\sigma$-finite length, Publ. Mat. 40 (1996), no. 1, 195–204. MR 1397014, DOI 10.5565/PUBLMAT_{4}0196_{1}2
- Pertti Mattila, Mark S. Melnikov, and Joan Verdera, The Cauchy integral, analytic capacity, and uniform rectifiability, Ann. of Math. (2) 144 (1996), no. 1, 127–136. MR 1405945, DOI 10.2307/2118585
- Yuval Peres, Wilhelm Schlag, and Boris Solomyak, Sixty years of Bernoulli convolutions, Fractal geometry and stochastics, II (Greifswald/Koserow, 1998) Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39–65. MR 1785620
- Dorothy Browne Shaffer, Radii of starlikeness and convexity for special classes of analytic functions, J. Math. Anal. Appl. 45 (1974), 73–80. MR 330435, DOI 10.1016/0022-247X(74)90121-8
- D. C. Spencer, On finitely mean valent functions. II, Trans. Amer. Math. Soc. 48 (1940), 418–435. MR 2603, DOI 10.1090/S0002-9947-1940-0002603-1
- Xavier Tolsa, $L^2$-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), no. 2, 269–304. MR 1695200, DOI 10.1215/S0012-7094-99-09808-3
- Xavier Tolsa, Bilipschitz maps, analytic capacity, and the Cauchy integral, Ann. of Math. (2) 162 (2005), no. 3, 1243–1304. MR 2179730, DOI 10.4007/annals.2005.162.1243
- Xavier Tolsa, Growth estimates for Cauchy integrals of measures and rectifiability, Geom. Funct. Anal. 17 (2007), no. 2, 605–643. MR 2322495, DOI 10.1007/s00039-007-0598-7
- Xavier Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, Progress in Mathematics, vol. 307, Birkhäuser/Springer, Cham, 2014. MR 3154530, DOI 10.1007/978-3-319-00596-6
- P. D. Tuan and V. V. Anh, Radii of starlikeness and convexity for certain classes of analytic functions, J. Math. Anal. Appl. 64 (1978), no. 1, 146–158. MR 480978, DOI 10.1016/0022-247X(78)90027-6
Additional Information
- Xin-Han Dong
- Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
- MR Author ID: 240828
- Email: xhdong@hunnu.edu.cn
- Ka-Sing Lau
- Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
- MR Author ID: 190087
- Email: kslau@math.cuhk.edu.hk
- Hai-Hua Wu
- Affiliation: College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China – and – Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, People’s Republic of China
- Email: hunaniwa@163.com
- Received by editor(s): December 12, 2014
- Received by editor(s) in revised form: July 16, 2015
- Published electronically: December 7, 2016
- Additional Notes: This research was supported in part by the NNSF of China (No. 11571099), an HKRGC grant, SRFDP of Higher Education (No. 20134306110003), and Scientific Research Fund of Hunan Provincial Education Department (No. 14K057). The first author is the corresponding author
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 4817-4842
- MSC (2010): Primary 28A80; Secondary 30C55, 30E20
- DOI: https://doi.org/10.1090/tran/6819
- MathSciNet review: 3632551