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Ulyanov-type inequalities and generalized Liouville derivatives

Published online by Cambridge University Press:  11 February 2011

Sergey Tikhonov  and
Walter Trebels
Sergey Tikhonov
Affiliation:
ICREA and Centre de Recerca Matemàtica, Apartat 50, 08193 Bellaterra, Barcelona, Spain (stikhonov@crm.cat)
Walter Trebels
Affiliation:
Fachbereich Mathematik, AG AGF, TU Darmstadt, 64289 Darmstadt, Germany (trebels@mathematik.tu-darmstadt.de)
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Abstract

We study (p, q)-inequalities of Ulyanov type for moduli of smoothness of fractional order in the Lp and the Lp(ℝn) setting, p ≥ 1. In particular, we obtain estimates for the modulus of smoothness of a generalized Liouville derivative of a function via the modulus of smoothness of the function itself. We give examples showing the sharpness of these inequalities.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 141 , Issue 1 , February 2011 , pp. 205 - 224
Copyright
Copyright © Royal Society of Edinburgh 2011

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