It is known from P. Lévy, Théorie de l’addition des variables aléatoires. Monographies des Probabilités ; calcul des probabilités et ses applications 1, Paris (1937), that the sample paths of the Wiener process $W=\left(W_t\right)_{t \geq 0}$ satisfy almost surely the following Hölder condition, for sufficiently small $|t^{\prime}-t|$,
$$\left|W_{t^{\prime}}-W_t\right| \leq c \cdot \sqrt{\left|t^{\prime}-t\right| \log \left(1+\frac{1}{\left|t^{\prime}-t\right|}\right)}$$
I wonder if there is a generalisation of this result for brownian sheet/Wiener field ? More precisely, for $W=\left(W_{t_{1},...,t_{d}}\right)_{t_{1}\geq 0,...,t_{d}\geq 0}$ the $d-$fold Wiener field, do we have almost surely, for sufficiently small $\left\|(t^{'}_{1},...,t^{'}_{d})-(t_{1},...,t_{d})\right\|$,
$$\left|W_{t^{'}_{1},...,t^{'}_{d}}-W_{t_{1},...,t_{d}}\right| \leq c \cdot \sqrt{\left\|(t^{'}_{1},...,t^{'}_{d})-(t_{1},...,t_{d})\right\| \log \left(1+\frac{1}{\left\|(t^{'}_{1},...,t^{'}_{d})-(t_{1},...,t_{d})\right\|}\right)}$$
PS : Does this result hold in expectation too (even in the d=1 case) ? More precisely, do $$\sup_{t,t^{'}\in[0,1]^{d}}\frac{\left|W_{t^{'}_{1},...,t^{'}_{d}}-W_{t_{1},...,t_{d}}\right|}{\sqrt{\left\|(t^{'}_{1},...,t^{'}_{d})-(t_{1},...,t_{d})\right\| \log \left(1+\frac{1}{\left\|(t^{'}_{1},...,t^{'}_{d})-(t_{1},...,t_{d})\right\|}\right)}}$$ admits finite moments ?
Regarding the first question: this follows from the fact that $$ \mathrm E |W(t') - W(t)|^2 \le ||t'-t||, $$ see e.g. Lifshitz Gaussian Random Functions, p. 220.
Existence of all moments follows from the Fernique theorem. Loosely it can be formulated as follows: if $X$ is centered Gaussian, and $||\cdot||$ is a seminorm such that $||X||<\infty$ almost surely, then there exists some $\alpha>0$ such that $\mathrm E e^{\alpha ||X||^2} <\infty$. Here you can take $X = W$, $||x|| = \sup\limits_{t'\neq t}\frac{|x(t') - x(t)|}{\big(|t'-t| \log (1 + |t' -t|^{-1})\big)^{1/2}}$.