I did not find this question, so I hope that someone can help me.
I am trying to understand the following inclusion $$W^{1,2}_{p,loc}(\mathbb{R}^{d+1})\hookrightarrow C^{0,1}_{\alpha, loc}(\mathbb{R}^{d+1}),$$
where $p>d+2$ and $\alpha=1-\frac{d+2}{p}$.
So, we I would like to show that a function $u(t,x)$, with week time derivative and up to second order week derivatives (only in space) are $L^p$ bounded. Then the function $u$ belongs to the parabolic Holder space where the distance is defined as $d((t,x),(s,y)):=|t-s|^\frac{1}{2}+ \|x-y\|_e$.
I would like a reference for this result where the proof is sufficiently complete.
Moreover, this is a specific result but it holds in a generic setting about the order of week derivatives.