Hausdorff measures, capacities and compact composition operators

Abstract

It is shown that there exist analytic self-maps phi of the unit disc D inducing compact composition operators on the Hardy space H-p, 1 <= p < infinity such that the Hausdorff dimension of the set E-omega = {e(i theta) is an element of partial derivative D: vertical bar phi(e(i theta))vertical bar = 1} is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces D-alpha with 0 < alpha < 1. As a consequence, we deduce that there exist symbols. inducing compact composition operators on D-alpha such that the alpha-capacity of E-phi is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on D-alpha.