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

Abstract

For any given time lag $γ\in(0,1)$, we prove that the one-sided parabolic BMO space $\mathrm{BMO}^+(γ)$ coincides with the parabolic BMO space $\mathrm{PBMO}^-(γ)$ with equivalent norms, the parabolic Muckenhoupt class $A_{\infty}^+(γ)$ defined via the reverse Jensen inequality can be represented as the union of the parabolic Muckenhoupt classes $A_r^+(γ)$ with $r\in[1,\infty)$, and the parabolic reverse Hölder classes $\bigcup_{q\in(1,\infty]}RH_q^+$ coincide with the parabolic Muckenhoupt classes $\bigcup_{r\in[1,\infty)}A_r^+(γ)$, 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 $\mathrm{BMO}^+(γ)$. As applications, we obtain John--Nirenberg and exponential integrability characterizations of $\mathrm{BMO}^+(γ)$, prove that $\mathrm{BMO}^+(γ)$ is independent of the positive time lag, and identify its null space.