Parabolic BMO, Parabolic Muckenhoupt Classes, and Parabolic Reverse Hölder Classes with Time Lag

We prove that, for any given time lag $\gamma \in (0,1)$, \begin{itemize} \item the one-sided parabolic BMO space ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\gamma )$ coincides with the parabolic BMO space ${\mathrm{P}\mathrm{B}\mathrm{M}\mathrm{O}}^{-}(\gamma )$, i.e., ${\mathrm{P}\mathrm{B}\mathrm{M}\mathrm{O}}^{-}(\gamma )={\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\gamma )$ with equivalent norms; \item the parabolic Muckenhoupt class $A_{\mathrm{\infty }}^{+}(\gamma )$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^{+}(\gamma )$ with $r\in [1,\mathrm{\infty })$, i.e., $A_{\mathrm{\infty }}^{+}(\gamma )={\bigcup }_{r\in [1,\mathrm{\infty })}{A}_r^{+}(\gamma )$; \item the parabolic reverse Hölder classes ${\bigcup }_{q\in (1,\mathrm{\infty }]}R{H}_q^{+}$ coincide with the parabolic Muckenhoupt classes ${\bigcup }_{r\in [1,\mathrm{\infty })}{A}_r^{+}(\gamma )$, i.e., ${\bigcup }_{q\in (1,\mathrm{\infty }]}R{H}_q^{+}={\bigcup }_{r\in [1,\mathrm{\infty })}{A}_r^{+}(\gamma )$, \end{itemize} and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016), 289–299]. The main ingredients are a uniform parabolic space-time shifting property for parabolic reverse Hölder weights and a one-sided stopping-time argument, which yields a new parabolic John–Nirenberg inequality for ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\gamma )$. As consequences, we obtain John–Nirenberg and exponential integrability characterizations of ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\gamma )$, prove that ${\mathrm{B}\mathrm{M}\mathrm{O}}^{+}(\gamma )$ is independent of the positive time lag, and identify its null space.