×

zbMATH — the first resource for mathematics

Large sets of zero analytic capacity. (English) Zbl 1062.30025
The authors consider a Cantor type set in the unit square and prove that it has zero analytic capacity but it does not have \(\sigma\)-finite one-dimensional Hausdorff measure. The first example of such a Cantor set was given by L. D. Ivanov [On sets of analytic capacity zero, in: Linear and Complex Analysis Problem Book 3, Part II (V.P. Khavin, S.V. Kruschev, N.K. Nikolskii, ed.), Lecture Notes Math. 1043, 498–501 (1984; Zbl 0545.30038)]. The proof in the paper under review is based on some estimates for harmonic measure and Green function due to P. W. Jones [Lecture Notes Math. 1384, 24–68 (1989; Zbl 0675.30029)] and a stopping-time argument. The result is related to a conjecture of P. Mattila [Publ. Math., Barc. 40, No. 1, 195–204 (1996; Zbl 0888.30026)] on the characterization of Cantor sets with zero analytic capacity.

MSC:
30C85 Capacity and harmonic measure in the complex plane
28A75 Length, area, volume, other geometric measure theory
28A78 Hausdorff and packing measures
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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. · Zbl 0745.42008
[2] 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. · Zbl 0919.28004 · doi:10.1007/BF02312776 · doi.org
[3] John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. · Zbl 0208.09803
[4] John Garnett, Analytic capacity and measure, Lecture Notes in Mathematics, Vol. 297, Springer-Verlag, Berlin-New York, 1972. · Zbl 0253.30014
[5] V. P. Havin, S. V. Hruščëv, and N. K. Nikol\(^{\prime}\)skiĭ , Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems.
[6] 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. · doi:10.1007/BFb0086793 · doi.org
[7] Pertti Mattila, On the analytic capacity and curvature of some Cantor sets with non-\?-finite length, Publ. Mat. 40 (1996), no. 1, 195 – 204. · Zbl 0888.30026 · doi:10.5565/PUBLMAT_40196_12 · doi.org
[8] 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. · Zbl 0897.42007 · doi:10.2307/2118585 · doi.org
[9] M. S. Mel\(^{\prime}\)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. · Zbl 0840.30008 · doi:10.1070/SM1995v186n06ABEH000045 · doi.org
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.