We introduce a class of weights related to the regularity theory of nonlinear parabolic
partial differential equations. In particular, we investigate connections of the
parabolic Muckenhoupt weights to the parabolic BMO. The parabolic Muckenhoupt
weights need not be doubling and they may grow arbitrarily fast in the time variable.
Our main result characterizes them through weak- and strong-type weighted norm
inequalities for forward-in-time maximal operators. In addition, we prove a
Jones-type factorization result for the parabolic Muckenhoupt weights and a
Coifman–Rochberg-type characterization of the parabolic BMO through maximal
functions. Connections and applications to the doubly nonlinear parabolic PDE are
also discussed.