Domains with globally exponentially integrable parabolic forward-in-time BMO

We characterize those open sets of the space time in which parabolic forward-in-time BMO functions are in a certain forward-in-time exponential integrability class. The characterization holds under qualitative connectivity assumptions on the domain and is formulated in terms of a quantitative growth bound on a forward-in-time version of the classical quasihyperbolic distance.