  We study if the parabolic forward-in-time maximal operator is bounded on
parabolic BMO. It turns out that for non-negative functions the answer is
positive, but the behaviour of sign changing functions is more delicate. The
class parabolic BMO and the forward-in-time maximal operator originate from the
regularity theory of nonlinear parabolic partial differential equations. In
addition to that context, we also study the question in dimension one.
