Parabolic BMO Spaces, Muckenhoupt Weights, and Reverse Hölder Classes with Time Lag: Equivalence and Characterizations

Abstract

For any given time lag $\gamma \in (0,1)$, we prove that the one-sided parabolic BMO space ${\mathrm{BMO}}^{+}(\gamma )$ coincides with the parabolic BMO space ${\mathrm{PBMO}}^{-}(\gamma )$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty }^{+}(\gamma )$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^{+}(\gamma )$ with $r\in [1,\infty )$, and the parabolic reverse Hölder classes ${\bigcup }_{q\in (1,\infty ]}R{H}_q^{+}$ coincide with the parabolic Muckenhoupt classes ${\bigcup }_{r\in [1,\infty )}{A}_r^{+}(\gamma )$, and hence give affirmative answers to Questions 4.5 and 4.6 posed by Kinnunen and Saari [Nonlinear Anal. 131 (2016)]. To show them, we establish the uniform parabolic space-time shifting property for parabolic reverse Hölder weights, and develop the one-sided stopping time argument which yields a new parabolic John--Nirenberg inequality for ${\mathrm{BMO}}^{+}(\gamma )$. As applications, we obtain John--Nirenberg and exponential integrability characterizations of ${\mathrm{BMO}}^{+}(\gamma )$, prove that ${\mathrm{BMO}}^{+}(\gamma )$ is independent of the positive time lag, and identify its null space.