Compactness of commutators of rough singular integrals
Aapo Laukkarinen, Jaakko Sinko
TL;DR
The paper characterizes the Bloom-type compactness of commutators [b,TΩ] for rough homogeneous kernels with Ω ∈ L^q(S^{d-1}). It proves that [b,TΩ]: L^u(μ)→L^v(λ) is compact if and only if b ∈ VMO^α_ν(ℝ^d), with α/d = 1/u - 1/v and ν^{1+α/d} = μ^{1/u} λ^{-1/v}, and it develops a density-based argument to extend results from Lipschitz Ω to general Ω. Additionally, it provides a matrix-weighted analogue: for matrix weights W ∈ A_p and Ω ∈ L^{q+ε}, the commutator is compact on L^p(W) when b ∈ VMO, via a matrix-weighted Kolmogorov–Riesz framework. The work extends known Calderón–Zygmund compactness results to rough kernels and offers a robust toolkit (including density arguments and factorisation) for two-weight and matrix-weighted settings.
Abstract
We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators $T_Ω$ that are associated with a kernel $Ω\in L^q(\mathbb{S}^{d-1})$. We establish a characterisation of compactness of the commutator $[b,T_Ω]$ in terms of the function $b$ belonging to a suitable space of functions with vanishing mean oscillation. Our results expand upon the previous compactness characterisations for Calderón-Zygmund operators. Additionally, we prove a matrix-weighted compactness result for $[b,T_Ω]$ by applying the so-called matrix-weighted Kolmogorov-Riesz theorem.