modulus of continuity of Brownian Motion Is there any estimate for the following quantity
$$
E\left(\sup_{\substack{0 \leq s,t \leq1 \\  |t-s| < \delta}} \left|W_t - W_s\right|\right)
$$
for some small $\delta > 0$, where $W$ is a Brownian motion.
I'm hoping for some result of the form
$$
E\left(\sup_{\substack{0 \leq s,t \leq1\\  |t-s| < \delta}}\left|W_t - W_s\right|\right) \leq \delta^{\frac{1}{2}}.
$$
Thanks!

Actually, it turns out that there exists constants $c_1$ and $c_2$ such that 
$$c_1\left(\delta\log\frac 2\delta\right)^{1/2}\leqslant \mathbb E\left(\sup_{\substack{0 \leqslant  s,t \leqslant 1 \\  |t-s| < \delta}} \left|W_t - W_s\right|\right) \leqslant c_2\left(\delta\log\frac 2\delta\right)^{1/2}$$
(see Section 4 in the paper by Fischer and Nappo). Note that they also have good bounds for the $p$th moment  of the modulus of continuity of the Brownian motion.
Reference: Fischer, Markus(D-HDBG-A); Nappo, Giovanna(I-ROME)
On the moments of the modulus of continuity of Itô processes. (English summary)
Stoch. Anal. Appl. 28 (2010), no. 1, 103–122. 
