On the smallness of mean oscillations on metric-measure spaces and applications

It will be established that the mean oscillation of a function on a metric-measure space $X\times Y$ will be small if its mean oscillation on $X$ is small and some simple information on its (partial $Y$) upper-gradient is given. Applications to the regularity and global existence of bounded solutions to strongly coupled elliptic/parabolic systems on thin domains are also considered.