My question concern some continuity properties of the Wiener field (aslo called sometime Brownian sheet), you can find a definition here : https://encyclopediaofmath.org/wiki/Wiener_field
Let consider $W$ a $d$-fold Wiener field over $[0,1]^{d}$ and let denote, for a Borel subset $A$ of $[0,1]^{d}$, $W(A)=\int_{A}dW$. If I understand well the definition, $W(.)$ can be thought as a random function over bounded Borel set such that $W(A)$ has a centered normal distribution and $\mathbb{E}(W(A)W(B))$ is equal to the Lebesgue measure of $A\cap B$, denoted $L(A\cap B)$.
Again, if I get thing well, for $A$ and $B$ are two Borel subsets of $[0,1]^{d}$ such that the Lebesgue measure of the symmetric difference $A\Delta B$ is small, $|W(A)-W(B)|=|W(A\Delta B)|$ will be small with high probability. It seems obvious that for sufficiently large $C$, with high probability we have $|W(A)-W(B)|\leq C L(A\Delta B)^{\alpha}$ for any $\alpha<1/2$. We can even say $|W(A)-W(B)|\leq C (L(A\Delta B))^{1/2}$.
My question is does this hold uniformly (up to paying some additional control) ? More precisely, can we say that we have, for sufficiently large $C$, with high probability, $$\sup\limits_{A,B\subset[0,1]^{d},\text{ Borel sets}}\frac{|W(A)-W(B)|}{L(A\Delta B)^{\alpha}}\leq C\text{ , for any } \alpha<1/2$$ or even, $$\sup\limits_{A,B\subset[0,1]^{d},\text{ Borel sets}}\frac{|W(A)-W(B)|}{(L(A\Delta B)\log(1+1/L(A\Delta B))^{1/2}}\leq C$$