 Inhomogeneous oscillation decay implies Hölder continuity

If the essential oscillation on concentric balls obeys $\omega (R/2)\le \theta \omega (R)+A{R}^{\alpha }$ with $0<\theta <1$ and $\alpha >0$, dyadic iteration gives $\omega (r)\le C{r}^{\mu }$ for every $0<\mu <\min \{\alpha ,-{\log }_2\theta \}$. The resulting decay gives a Hölder continuous function. Taking the exponent strictly below both decay rates avoids a logarithmic factor when the rates coincide.