Matrix-weighted little BMO spaces in two parameters

In this paper we set up a theory of two-matrix weighted little BMO in two parameters. We prove that being a member of this class is equivalent to belonging uniformly in each variable to two-matrix weighted (one-parameter) BMO, a class studied extensively by J. Isralowitz, S. Pott, S. Treil and others. Using this equivalence, we deduce lower and upper bounds in terms of the two-matrix weighted little BMO norm of the symbol for the norm of commutators with Journé operators.