Martingale-coboundary representation for stationary random fields

We prove a martingale-coboundary representation for random fields with a completely commuting filtration. For random variables in L-2, we present a necessary and sufficient condition which is a generalization of Heyde's condition for one-dimensional processes from 1975. For L-p spaces with 2 <= p < infinity 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. A new sufficient condition is presented which for dimension one improves Gordin's condition from 1969. In application, new weak invariance principle and estimates of large deviations are found.