Let $\Omega\subset\mathbb{R}^n$ be a bounded domain of class $C^1$ and $\gamma\in (0,1]$.
I am only interested in Hoelder spaces of form $C^{0,\gamma}(\overline{\Omega})$, i.e. I am not interested in whether the functions in the space have Hoelder-continuous derivatives or not.
Thanks a lot for your help.
Let $|u|_{0,\gamma}=\displaystyle\sup_{x,y\in\overline{\Omega}, x\neq y}\frac{|u(x)-u(y)|}{|x-y|^\gamma}$. Then the norm on $C^{0,\gamma}(\overline{\Omega})$ is given by $$\|u\|_{0,\gamma}=\|u\|_{C(\overline{\Omega})}+|u|_{0,\gamma}$$
If $|\Omega|$ denote the measure of $\Omega$, we have that for $p\in [1,\infty)$ \begin{eqnarray} \Big(\int_\Omega |u|^p\Big)^{\frac{1}{p}} &\leq& |\Omega|^\frac{1}{p}\|u\|_{C(\overline{\Omega})} \nonumber \\ &\leq& |\Omega|^{\frac{1}{p}}\|u\|_{C(\overline{\Omega})}+|\Omega|^\frac{1}{p}|u|_{0,\gamma} \nonumber \\ &=& |\Omega|^\frac{1}{p} \|u\|_{0,\gamma} \end{eqnarray}
The case $p=\infty$ is trivial.