Generalized Hölder continuity and oscillation functions

We study a notion of generalized H\"older continuity for functions on \(\mathbb{R}^d\). We show that for any bounded function \(f\) of bounded support and any \(r>0\), the \(r\)-oscillation of \(f\) defined as \(osc_r f (x):= \sup_{B_r(x)} f - \inf_{B_r(x)} f\) is automatically generalized H...

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Bibliographic Details
Published inarXiv.org
Main Author Imre Péter Tóth
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 11.10.2018
Subjects
Continuity (mathematics)
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Summary:We study a notion of generalized H\"older continuity for functions on \(\mathbb{R}^d\). We show that for any bounded function \(f\) of bounded support and any \(r>0\), the \(r\)-oscillation of \(f\) defined as \(osc_r f (x):= \sup_{B_r(x)} f - \inf_{B_r(x)} f\) is automatically generalized H\"older continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.
ISSN:2331-8422