Does Log-Lipschitz regularity imply Hölder continuity? A function is Log-Lipschitz if there exists a constant $C$ such that
\begin{equation}
|u(x) - u(y)| \le C|x-y| \log|x-y|
\end{equation}
Is a Log-Lipschitz function $C^{0,\alpha}$ for any $\alpha \in (0,1)
$(Hölder continuous)?
If you need, assume hypothesis. Thank you.

Yes, it is -- assuming that you think and act locally. Think in terms of moduli of continuity $\omega$, i.e., functions such that $|u(x)-u(y)|\le \omega(|x-y|)$. The function $\delta\log (1/\delta)$ is smaller than $\delta^{\alpha}$ ($\alpha<1$) near $0$.
Edit: For $|x-y| < 1$ it is clear that $\log|x-y| < 0$. Commonly this is fixed by adding the modulus: $|\log |x-y||$. In the notation of moduli of continuity the issue is resolved by writing $\log(1/\delta) = - \log(\delta)$, where $\delta$ is small.
