  We prove a martingale-coboundary representation for random fields with a
completely commuting filtration. For random variables in L2 we present a
necessary and sufficient condition which is a generalization of Heyde's
condition for one dimensional processes from 1975. For Lp spaces with 2 \leq p
< \infty we give a necessary and sufficient condition which extends Volny's
result from 1993 to random fields and improves condition of El Machkouri and
Giraudo from 2016 (arXiv:1410.3062). In application, new weak invariance
principle and estimates of large deviations are found.
