Generalized Hölder Continuity and Oscillation Functions

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