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Hausdorff measures, capacities and compact composition operators

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Gallardo Gutiérrez, Eva A. and González, María J. (2006) Hausdorff measures, capacities and compact composition operators. Mathematische Zeitschrift, 253 (1). pp. 63-74. ISSN 0025-5874

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Official URL: http://link.springer.com/content/pdf/10.1007%2Fs00209-005-0878-6.pdf


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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.


Item Type:Article
Uncontrolled Keywords:composition operator; Weighted Dirichlet spaces; Hausdorff measures
Subjects:Sciences > Mathematics > Mathematical analysis
ID Code:21104
Deposited On:26 Apr 2013 08:32
Last Modified:10 Aug 2018 11:02

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