Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cauchy transforms of self-similar measures: Starlikeness and univalence


Authors: Xin-Han Dong, Ka-Sing Lau and Hai-Hua Wu
Journal: Trans. Amer. Math. Soc. 369 (2017), 4817-4842
MSC (2010): Primary 28A80; Secondary 30C55, 30E20
DOI: https://doi.org/10.1090/tran/6819
Published electronically: December 7, 2016
MathSciNet review: 3632551
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A] 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
  • [C] Vasilis Chousionis, Singular integrals on Sierpinski gaskets, Publ. Mat. 53 (2009), no. 1, 245-256. MR 2474123, https://doi.org/10.5565/PUBLMAT_53109_11
  • [CMT] 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, https://doi.org/10.1007/s00526-014-0766-1
  • [CMa] 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, https://doi.org/10.1515/crelle-2012-0078
  • [CU] Vasileios Chousionis and Mariusz Urbański, Homogeneous kernels and self-similar sets, Indiana Univ. Math. J. 64 (2015), no. 2, 411-431. MR 3344433, https://doi.org/10.1512/iumj.2015.64.5491
  • [Ch] 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
  • [CMR] 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
  • [Da] 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, https://doi.org/10.4171/RMI/242
  • [D] 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
  • [DL1] 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, https://doi.org/10.1016/S0022-1236(02)00069-1
  • [DL2] 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
  • [DL3] 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, Inc., Boston, MA, 2010, pp. 283-294. MR 2743000, https://doi.org/10.1007/978-0-8176-4888-6_18
  • [DL4] X. H. Dong and K. S. Lau, Cauchy Transform on Sierpinski Gasket: fractal behavior at the boundary, submitted.
  • [DLL] Xin-Han Dong, Ka-Sing Lau, and Jing-Cheng Liu, Cantor boundary behavior of analytic functions, Adv. Math. 232 (2013), 543-570. MR 2989993, https://doi.org/10.1016/j.aim.2012.09.021
  • [Du] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [F] Kenneth Falconer, Fractal geometry: Mathematical foundations and applications, 2nd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2003. MR 2118797
  • [G1] John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. MR 0276456
  • [G2] John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR 0454006
  • [H] D. J. Hallenbeck, Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math. Soc. 192 (1974), 285-292. MR 0338338
  • [Hay] W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Mathematics, vol. 110, Cambridge University Press, Cambridge, 1994. MR 1310776
  • [Hut] John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. MR 625600, https://doi.org/10.1512/iumj.1981.30.30055
  • [L] Ka-Sing Lau, Dimension of a family of singular Bernoulli convolutions, J. Funct. Anal. 116 (1993), no. 2, 335-358. MR 1239075, https://doi.org/10.1006/jfan.1993.1116
  • [LN] Ka-Sing Lau and Sze-Man Ngai, Multifractal measures and a weak separation condition, Adv. Math. 141 (1999), no. 1, 45-96. MR 1667146, https://doi.org/10.1006/aima.1998.1773
  • [LDP] 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, https://doi.org/10.1016/j.jmaa.2013.06.059
  • [LSV] 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
  • [Mac] Thomas H. MacGregor, The radius of convexity for starlike functions of order $ {1\over 2}$, Proc. Amer. Math. Soc. 14 (1963), 71-76. MR 0150282
  • [MTV] Joan Mateu, Xavier Tolsa, and Joan Verdera, The planar Cantor sets of zero analytic capacity and the local $ T(b)$-theorem, J. Amer. Math. Soc. 16 (2003), no. 1, 19-28 (electronic). MR 1937197, https://doi.org/10.1090/S0894-0347-02-00401-0
  • [Ma1] Pertti Mattila, Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. MR 1333890
  • [Ma2] Pertti Mattila, Smooth maps, null-sets for integralgeometric measure and analytic capacity, Ann. of Math. (2) 123 (1986), no. 2, 303-309. MR 835764, https://doi.org/10.2307/1971273
  • [Ma3] Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-$ \sigma $-finite length, Publ. Mat. 40 (1996), no. 1, 195-204. MR 1397014, https://doi.org/10.5565/PUBLMAT_40196_12
  • [MMV] 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, https://doi.org/10.2307/2118585
  • [PPS] 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
  • [Sh] Dorothy Browne Shaffer, Radii of starlikeness and convexity for special classes of analytic functions, J. Math. Anal. Appl. 45 (1974), 73-80. MR 0330435
  • [Sp] D. C. Spencer, On finitely mean valent functions. II, Trans. Amer. Math. Soc. 48 (1940), 418-435. MR 0002603
  • [To1] Xavier Tolsa, $ L^2$-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), no. 2, 269-304. MR 1695200, https://doi.org/10.1215/S0012-7094-99-09808-3
  • [To2] Xavier Tolsa, Bilipschitz maps, analytic capacity, and the Cauchy integral, Ann. of Math. (2) 162 (2005), no. 3, 1243-1304. MR 2179730, https://doi.org/10.4007/annals.2005.162.1243
  • [To3] Xavier Tolsa, Growth estimates for Cauchy integrals of measures and rectifiability, Geom. Funct. Anal. 17 (2007), no. 2, 605-643. MR 2322495, https://doi.org/10.1007/s00039-007-0598-7
  • [To4] Xavier Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, Progress in Mathematics, vol. 307, Birkhäuser/Springer, Cham, 2014. MR 3154530
  • [TA] 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 0480978

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 28A80, 30C55, 30E20

Retrieve articles in all journals with MSC (2010): 28A80, 30C55, 30E20


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
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
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

DOI: https://doi.org/10.1090/tran/6819
Keywords: Self-similar measure, Cauchy transform, radius of starlikeness, convexity, univalence
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
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society