Hausdorff measure and capacity associated with Cauchy potentials

In the paper the connection between the Hausdorff measure _h(E) of setsE and the analytic capacity(E), and also between _h(E) and the capacity⁺(E) generated by Cauchy potentials with nonnegative measures is studied. It is shown that if the integral 0t^–3h²(t)dt is divergent andh satisfies the regularity condition, then there exists a plane Cantor setE for which _h(E)>0, but⁺(E)=0. The proof is based on the estimate of⁺(E_n), whereE_n is the set appearing at thenth step in the construction of a plane Cantor set.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 923–934, June, 1998.I am grateful to M. S. Melnikov and X. Tolsa for information about the work [17] and for discussion of open problems.I wish to thank the Department of Mathematics of Uppsala University (Sweden) for the excellent conditions under which this research was begun, and to Professor M. Essén for the hospitality and support of this undertaking. I am grateful also to Professor P. Mattila and Professor S. Ya. Khavinson for very useful discussions of these problems.Research partially supported by the Royal Swedish Academy of Sciences, International Science Foundation (Grant MB 6000) and by Ministry of Education of Russia (Grant 95-0-1.7-52)This research was partially supported by the Royal Swedish Academy of Sciences, by the International Science Foundation under grant No. MB 6000 and by the Ministry of Education of Russia under grant No. 95-0-1.7-52.