Generalized Hölder continuity and oscillation functions

Abstract

We study a notion of generalized Hölder 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 $os{c}_{r}f(x):={\sup }_{B_r(x)}f-{\inf }_{B_r(x)}f$ 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.

3rd version: a discussion of the results added; a minor gap in the proof fixed; some typos corrected. The results are unchanged