We know the paths of Brownian motion are a.s. hölder continous with $\alpha \in (0,\frac{1}{2})$ that is a.s. $$|W(t)-W(s)|\leq C(\omega)|t-s|^{\alpha} $$ for $t,s\geq0$. This result is tight. What about a similiar result in $L^p$ norms ($p\geq 1$)? $$\mathbb{E}[|W(t)-W(s)|^p]^{\frac{1}{p}}\leq C|t-s|^{\alpha}$$ In my textbook this case is not covered altough this seems to me like a nice addition to hölder continuity of Brownian motion if it is true.
In mathoverflow it is claimed to be true for $\alpha \in (0,\frac{1}{2})$ but I have some trouble understanding the proof. Also we still don't know whether it's tight. https://mathoverflow.net/questions/304238/moments-of-the-h%C3%B6lder-norm-of-brownian-process
I'm especially interested in case $p=2$ and I'd appreciate some tips for literature about this.