It is known that if $(S_i= \sum_{j \leqslant i }X_i, \mathcal F_i)$ is a martingale,
then for each
$
\beta>1$, $\delta\in (0,\beta-1)$ and $\lambda>0$, and each integer $N \geqslant 1$, the inequality
$$\tag{*}
\mu\left\{\max_{1\leqslant i\leqslant N}|S_i|>\beta\lambda\right\}
\leqslant \frac{\delta^2}{(\beta-\delta-1)^2}\mu\left\{\max_{1\leqslant i\leqslant N}|S_i|>\lambda \right\}+
\\
+\mu\left\{ \sum_{i=1}^N \mathbb E [X_i^2 \mid \mathcal F_{i-1}]>\delta^2\lambda^2\right\}
+\mu \left\{\max_{1\leqslant i\leqslant N}|X_i| > \delta\lambda^2\right\}
$$
takes place (see for instance Hall and Heyde Martingale limit theory and its applications,
page 28). One can deduce the upper bound for $\mathbb E\max_{1\leqslant i\leqslant N}|S_i|^p$
in Burkholder's inequality from (*).
Now, we are looking for multidimensional extensions. Below I present the so-called orthomartingales, but I will settle for any form of martingale difference random field.
Now, assume that $( \mathcal F_{n_1,n_2 })_{n_1,n_2 \in\mathbb Z }$ is a commuting filtration, that is, \begin{equation} \mathbb E\left[ \mathbb E[X\mid \mathcal F_{n_1,n_2} ]\mid \mathcal F_{n'_1,n'_2}\right] =\mathbb E\left[X\mid \mathcal F_{\min\{ n_1,n'_1\},\min\{ n_2,n'_2\}} \right] \end{equation} for each integrable random variable $X$ and each $(n_1,n_2), (n'_1,n'_2)$. We denote $\mathcal F_{i,\infty}$ (respectively $\mathcal F_{\infty,j}$) the $ \sigma$-algebra generated by $\bigcup_{j\in \mathbb Z}\mathcal F_{i,j}$ (resp. $\bigcup_{i\in \mathbb Z}\mathcal F_{i,j}$ ).
Assume that the random field $(X_{i,j})_{(i,j)\in\mathbb Z^2}$ is adapted to the filtration $( \mathcal F_{n_1,n_2 })_{n_1,n_2 \in\mathbb Z }$ an that it satisfies $ \mathbb E[X_{i,j} \mid \mathcal F_{i-1,\infty}]=X_{i-1,j}$ and $ \mathbb E[X_{i,j} \mid \mathcal F_{\infty,j-1}]=X_{i,j-1}$. The process $(\sum_{i_1=1}^{n_2}\sum_{i_2=1}^{n_2}X_{i_1,i_2} )_{n_1,n_2\in\mathbb N^2 }$ is called an orthomartingale, see Khosnevisian's book Multiparameter processes.
Question: Is there (or is it possible to establish) an inequality in the spirit of (*) for orthomartingale random fields?
Such an inequality would allow to derive a moment bound for $\max_{i\leqslant n_1,j\leqslant n_2}|S_{i,j}| ^p$. This can be derived by an application of Doob's inequality and Burkholder's one dimensional inequality.
I am not sure that (*) holds for submartingales; therefore we cannot apply it directly to $\max_{1\leqslant j\leqslant n_2}| S_{i,j}|$ for a fixed $i\leqslant n_1$.
The main difficulty is that we do not have a useful total order on $\mathbb N^2$. Also, it seems difficult to identify what would play the role of the quadratic variances for $d=1$.
