I'm currently studying for a course about Mean Field Theory and other scaling limits topics in probability.
An instrument that has been introduced is the Sobolev-Slobodeckij space $W^{\alpha,p}(0,T;\mathbb{R}^d)$ of all $L^p$ functions such that $$[f]_{W^{\alpha,p}}:=\int_0^T\int_0^T \frac{|f(t)-f(s)|}{|t-s|^{1+\alpha p}}dtds<\infty,$$ endowed with the norm $\|f\|_{W^{\alpha,p}}:=\|f\|_{L^p}+[f]_{W^{\alpha,p}}.$
It was told us that there is an inclusion: $$W^{\alpha,p}(0,T;\mathbb{R}^d)\hookrightarrow C^\varepsilon([0,T];\mathbb{R}^d) \ \ \text{if} \ \ (\alpha-\varepsilon)p>1,$$ and that there is a constant $C_{\alpha,\varepsilon,p}$ such that $$[f]_{C^\varepsilon}\le C_{\alpha,\varepsilon,p}\|f\|_{W^{\alpha,p}}.$$
Now, I tried to look for something about this but I couldn't find any reference, in particular I was willing to understand if the last inequality is true also if we put $[f]_{W^{\alpha,p}}$ instead of $\|f\|_{W^{\alpha,p}}.$
Do you know any article/book where I can read about this topic? Thanks to everybody.