Sharp Weighted Endpoint and Strong Estimates for Commutators of Rough Singular Integrals under the Log-Dini Condition

Abstract

In this paper, we establish optimal weighted norm inequalities for commutators of singular integral operators with rough kernels. While classical Calder\'{o}n-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.