Generalized Hölder Continuity and Oscillation Functions

  • Imre Péter TóthEmail author


We study a notion of generalized Hölder continuity for functions on 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 Hölder continuous, and we give an estimate for the appropriate (semi)norm. This is motivated by applications in the theory of dynamical systems.


Hölder continuity Function oscillation Regularisation Supremum smoothing 

Mathematics Subject Classification (2010)

28A75 37D99 



This research was supported by Hungarian National Research, Development and Innovation Office grants No. K 104745 and K 123782, and the Stiftung Aktion Österreich-Ungarn, project OMAA-92öu6. The author is grateful to Péter Bálint, Péter Nándori and Domokos Szász for the illuminating discussions on the problem, and to two anonymous referees for carefully checking the manuscript.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.MTA-BME Stochastics Research GroupBudapestHungary
  2. 2.Department of StochasticsBudapest University of Technology and EconomicsBudapestHungary

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