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Let $u \in C^{\alpha}(\Omega)$ and $B_R(x) \subset \Omega$. How can I see that \begin{equation} |u(x) - u_{B_R(x)}| \le (2R)^{\alpha} \| u\|_{C^\alpha(\Omega)}. \end{equation} where $u_{B_R(x)}= \dfrac{1}{|B_R(x)|}|u|$.

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    It is still not correct. As I said before, you are missing an integral in the definition of $u_{B_R(x)}$, and you have not defined the meaning of $C^\alpha$, the $C^\alpha$-(semi)norm and the range of $\alpha$. Anyway, assuming my interpretation above is correct, you should get the result by simply integrating the Hölder inequality over the ball.2012-10-25

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