On -convex functions of order with -fold symmetry.
We introduce a sequence of Hankel style operators , k = 1,2,3,..., which act on the Bergman space of the unit disk. These operators are intermediate between the classical big and small Hankel operators. We study the boundedness and Schatten-von Neumann properties of the and show, among other things, that are cut-off at 1/k. Recall that the big Hankel operator is cut-off at 1 and the small Hankel operator at 0.
In this paper we survey some recent results in connection with the so called Painlevé's problem and the semiadditivity of analytic capacity γ. In particular, we give the detailed proof of the semiadditivity of the capacity γ+, and we show almost completely all the arguments for the proof of the comparability between γ and γ+.