# Doubt about an estimate in Hölder spaces.

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!