Parabolic BMO Spaces, Muckenhoupt Weights, and Reverse H\"older 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 H\"older classes $\bigcup_{q\in(1,\infty]}RH_q^+$ 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 H\"older 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.