并矢点态控制与中值振荡分解的注记
Remark on dyadic pointwise domination and median oscillation decomposition
摘要：
在本文中，我们做了以下工作：a）利用Lacey最近的技巧，我们给出了Conde-Alonso和Rey控制定理的另一种证明，该定理指出每个任意复杂度的正并元算子都被一个零复杂度的正并元算子逐点控制：\[ \sum_{S\in\mathcal{S}} \langle f \rangle^μ_{S^{(k)}} 1_S\lesssim (k+1) \sum_{S'\in\mathcal{S}'} \langle f \rangle^μ_{S'} 1_{S'}. \] b）通过遵循中值和平均振荡之间的类似，我们将Lerner的局部中值振荡分解扩展到任意（可能非加倍）测度：\[\lvert f-m(f,\hat{S_0})\rvert 1_{S_0}\lesssim \sum_{S\in\mathcal{S}} (ω_λ(f;S)+\lvert m(f,S)-m(f,\hat{S}) \rvert )1_S.\]这可以被看作是适应并矢（鞅）BMO的中值振荡分解。作为应用程序的分解，我们给出了另一种证明并矢（鞅）约翰-尼伦伯格不等式，和莱西的控制定理，其中规定，每个鞅变换是逐点主导的正并矢运营商的复杂性为零。
英文摘要：
In this note, we do the following: a) By using Lacey's recent technique, we give an alternative proof for Conde-Alonso and Rey's domination theorem, which states that each positive dyadic operator of arbitrary complexity is pointwise dominated by a positive dyadic operator of zero complexity: \[ \sum_{S\in\mathcal{S}} \langle f \rangle^μ_{S^{(k)}} 1_S\lesssim (k+1) \sum_{S'\in\mathcal{S}'} \langle f \rangle^μ_{S'} 1_{S'}. \] b) By following the analogue between median and mean oscillation, we extend Lerner's local median oscillation decomposition to arbitrary (possibly non-doubling) measures: \[\lvert f-m(f,\hat{S_0})\rvert 1_{S_0}\lesssim \sum_{S\in\mathcal{S}} (ω_λ(f;S)+\lvert m(f,S)-m(f,\hat{S}) \rvert )1_S.\] This can be viewed as a median oscillation decomposition adapted to the dyadic (martingale) BMO. As an application of the decomposition, we give an alternative proof for the dyadic (martingale) John--Nirenberg inequality, and for Lacey's domination theorem, which states that each martingale transform is pointwise dominated by a positive dyadic operator of complexity zero.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
