带时滞的抛物型BMO空间、抛物型Muckenhoupt类与抛物型反向Hölder类
Parabolic BMO Spaces, Muckenhoupt Weights, and Reverse Hölder Classes with Time Lag: Equivalence and Characterizations
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中文总结 AI 辅助
本文研究带时滞的抛物型函数空间与相关类别的关系，证明三类抛物型空间的重合性，回答了Kinnunen与Saari的两个问题，给出BMO⁺(γ)的相关刻画
中文摘要 AI 辅助
我们证明，对于任意给定的时滞γ∈(0,1)，以下结论成立：其一，单侧抛物型BMO空间BMO⁺(γ)与抛物型BMO空间PBMO⁻(γ)重合，即PBMO⁻(γ)=BMO⁺(γ)，且二者的范数等价；其二，通过反向Jensen不等式定义的抛物型Muckenhoupt类A_∞⁺(γ)可表示为抛物型Muckenhoupt类A_r⁺(γ)（r∈[1,∞)）的并集，即A_∞⁺(γ)=∪_{r∈[1,∞)}A_r⁺(γ)；其三，抛物型反向Hölder类∪_{q∈(1,∞]}RH_q⁺与抛物型Muckenhoupt类∪_{r∈[1,∞)}A_r⁺(γ)重合，即∪_{q∈(1,∞]}RH_q⁺=∪_{r∈[1,∞)}A_r⁺(γ)，从而对Kinnunen与Saari在[Nonlinear Anal. 131 (2016), 289--299]中提出的问题4.5和问题4.6给出了肯定回答。主要研究工具包括抛物型反向Hölder权的一致抛物型时空平移性质，以及单侧停时论证，该论证导出了BMO⁺(γ)的一个新的抛物型John--Nirenberg不等式。作为推论，我们得到了BMO⁺(γ)的John--Nirenberg特征与指数可积性特征，证明了BMO⁺(γ)与正时滞无关，并确定了其零空间。
英文摘要
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.
