Large sets of zero analytic capacity

Authors:
John Garnett and Stan Yoshinobu

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3543-3548

MSC (2000):
Primary 30C20, 28A75

Published electronically:
June 13, 2001

MathSciNet review:
1860486

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We prove that certain Cantor sets with non-sigma-finite one- dimensional Hausdorff measure have zero analytic capacity.

**[C90]**Michael Christ,*Lectures on singular integral operators*, CBMS Regional Conference Series in Mathematics, vol. 77, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1104656****[E98]**V. Ya. Èĭderman,*Hausdorff measure and capacity associated with Cauchy potentials*, Mat. Zametki**63**(1998), no. 6, 923–934 (Russian, with Russian summary); English transl., Math. Notes**63**(1998), no. 5-6, 813–822. MR**1679225**, 10.1007/BF02312776**[G70]**John Garnett,*Positive length but zero analytic capacity*, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid.**26**(1970), 701. MR**0276456**, 10.1090/S0002-9939-1970-0276456-5**[G72]**John Garnett,*Analytic capacity and measure*, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. MR**0454006****[I84]**V. P. Havin, S. V. Hruščëv, and N. K. Nikol′skiĭ (eds.),*Linear and complex analysis problem book*, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR**734178****[J89]**Peter W. Jones,*Square functions, Cauchy integrals, analytic capacity, and harmonic measure*, Harmonic analysis and partial differential equations (El Escorial, 1987), Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 24–68. MR**1013815**, 10.1007/BFb0086793**[M96]**Pertti Mattila,*On the analytic capacity and curvature of some Cantor sets with non-𝜎-finite length*, Publ. Mat.**40**(1996), no. 1, 195–204. MR**1397014**, 10.5565/PUBLMAT_40196_12**[MMV96]**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**, 10.2307/2118585**[Me95]**M. S. Mel′nikov,*Analytic capacity: a discrete approach and the curvature of measure*, Mat. Sb.**186**(1995), no. 6, 57–76 (Russian, with Russian summary); English transl., Sb. Math.**186**(1995), no. 6, 827–846. MR**1349014**, 10.1070/SM1995v186n06ABEH000045

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
30C20,
28A75

Retrieve articles in all journals with MSC (2000): 30C20, 28A75

Additional Information

**John Garnett**

Affiliation:
Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095

Email:
jbg@math.ucla.edu

**Stan Yoshinobu**

Affiliation:
Department of Mathematics, University of Arizona, Tucson, Arizona 85721

Email:
syoshino@math.arizona.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06261-X

Received by editor(s):
April 7, 2000

Published electronically:
June 13, 2001

Additional Notes:
The authors were supported in part by NSF Grant DMS-0070782.

Communicated by:
Juha M. Heinonen

Article copyright:
© Copyright 2001
American Mathematical Society