turkmath.org

A. Volberg posed a problem about the existence of a compact set on the line with the corresponding "two-sided Hölder" Green function. This means that, given a point near the compact set, the Green function at this point has lower and upper estimates of Hölder type with the same exponent for all points. A solution of this problem is suggested by means of so-called weakly equilibrium Cantor-type sets. It is possible also to present Green functions that are "two-sided" with respect to other moduli of continuity.