  We investigate parabolic Muckenhoupt weights and functions of bounded mean
oscillation (BMO) related to nonlinear parabolic partial differential
equations. The main result gives a full characterization of weak and strong
type weighted norm inequalities for parabolic forward in time maximal
operators. In addition, we give a Jones type factorization result for the
parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of
the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt
weights and maximal functions.

Analysis of PDEs;Classical Analysis and ODEs