对数迪尼条件下粗糙奇异积分交换子的尖锐加权端点估计与强估计
Sharp Weighted Endpoint and Strong Estimates for Commutators of Rough Singular Integrals under the Log-Dini Condition
AI总结：
本文在对数迪尼条件下，为带粗糙核的奇异积分交换子建立了最优加权范数不等式，含$p=1$的尖锐端点估计，还通过反例证明该条件的最优性。
AI中文摘要：
本文针对带有粗糙核的奇异积分算子的交换子，建立了最优的加权范数不等式。经典的Calderón-Zygmund理论严重依赖点态梯度光滑性，而本文仅在核的$L^{1}(\boldsymbol{\textit{S}}^{n-1})$球面限制上假设严格更弱的对数迪尼（log-Dini）正则性条件。首先，证明这些粗糙交换子在整个Muckenhoupt权$w \in A_{p}$（$1<p<\infty$）范围内的加权勒贝格空间$L^{p}(w)$上是有界的。主要贡献是在临界值$p=1$处建立了尖锐的加权端点估计：对于任意权$w \in A_{1}$，该交换子满足带有精确$L\log L$对数损失的弱型不等式，在缺乏传统核正则性的情况下成功恢复了经典的光滑性行为。证明依赖于微局部分解的精细改进，结合涉及Orlicz平均的直接、局部稀疏控制框架。最后，通过构造基于缺项傅里叶级数振荡性质的严格反例，解决了最优性问题，该构造证明对数迪尼条件是尖锐的，确认无法放松对数正则性而不损失算子的基本有界性。
英文摘要：
In this paper, we establish optimal weighted norm inequalities for commutators of singular integral operators with rough kernels. While classical Calderón-Zygmund theory relies heavily on pointwise gradient smoothness, we operate under the strictly weaker log-Dini regularity condition assumed merely on the $L^{1}\left( \mathcal{S}^{n-1}\right) $ spherical restriction of the kernel. First, we prove that these rough commutators are bounded on the weighted Lebesgue spaces $L^{p}\left( w\right) $ for the full range of Muckenhoupt weights $w\in A_{p}$ $\left( 1<p<\infty \right) $. Our primary contribution establishes a sharp weighted endpoint estimate at the critical value $p=1$. For any weight $w\in A_{1}$, we demonstrate that the commutator satisfies a weak-type inequality with a precise $L\log L$ logarithmic loss, successfully recovering the classical smooth behavior in the absence of traditional kernel regularity. The proofs rely on a meticulous refinement of microlocal decompositions combined with a direct, localized sparse domination framework involving Orlicz averages. Finally, we settle the question of optimality by constructing a rigorous counterexample based on the oscillatory properties of lacunary Fourier series. This construction proves that the log-Dini condition is sharp, confirming that the logarithmic regularity cannot be relaxed without losing the operator's fundamental boundedness.
