BMO from dyadic BMO on the bidisc

We generalize to the bidisc a theorem of Garnett and Jones relating the space BMO of functions of bounded mean oscillation to its martingale counterpart, dyadic BMO. Namely, translation-averages of suitable families of dyadic BMO functions belong to BMO. As a corollary, we deduce a biparameter version of a theorem of Burgess Davis connecting the Hardy space H-1 to martingale H-1. We also prove the analogs of the theorem of Garnett and Jones in the one-parameter and biparameter VMO spaces of functions of vanishing mean oscillation.