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I am studying a paper about fluid dynamics and there is an estimate that involves norms in Hölder spaces that I do not know why it is true. Let be $R^2=n^2+z^2$, $\theta=\arctan\left(\dfrac{z}{n}\right)$, $P$ and $g$ functions that depends on time and $\theta$. Let be $\bar{u}=u-\left(RP-\dfrac{1}{n+1}\right)\phi$, being $\phi(R)=0$ if $R\le 1$ and $\phi(R)=0$ if $R\ge 2$. Then, the following estimate holds: $$ \left|\dfrac{\partial}{\partial z}(RP)\dfrac{\bar{u}\phi}{(n+1)^2}\right|_{C^{\alpha}}+\left|\dfrac{\partial\phi}{\partial z}\dfrac{RP\bar{u}}{(n+1)^2}\right|_{C^{\alpha}}\le C|P|_{C^{1,\alpha}([0,l])}\left|D\bar{u}\right|_{C^{\alpha}} $$ Thanks a lot!

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