A compact $T1$ theorem for Calderón-Zygmund operators associated with Zygmund dilations

TL;DR

This work develops a compact $T1$ theorem for Calderón--Zygmund operators linked to Zygmund dilations on $\mathbb{R}^3$, establishing compactness on weighted and unweighted $L^p$ spaces under full and partial kernel representations, weak compactness, and cancellation. The authors introduce a compact dyadic representation, expressing linear and bilinear CZ operators as averages of compact Zygmund dyadic shifts, enabling a robust path from dyadic models to operator compactness. They also extend the theory to bilinear operators, achieving compactness across the full range of exponents via dyadic representations and extrapolation/interpolation tools. A weighted theory is developed through Kolmogorov–Riesz criteria and extrapolation, with a precise distinction between strong and Zygmund weight classes. Overall, the paper deepens understanding of how dyadic decompositions and multi-parameter dilations govern compactness in singular integrals, offering a powerful framework for further weighted and multilinear extensions.

Abstract

We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_θ, δ_1, δ_{2, 3})$-Calderón-Zygmund operator $T$ associated with Zygmund dilations admits the compact full and partial kernel representations, and satisfies the weak compactness property and the cancellation condition, then $T$ can be extended to a compact operator on $L^p(w)$ whenever (i) $p \in (1, \infty)$, $w \in A_{p, \mathcal{R}}$, and $θ, δ_1, δ_{2, 3} \in (0, 1]$, or (ii) $p \in (1, \infty)$, $w \in A_{p, \mathcal{Z}}$, $θ= δ_1 = 1$, and $δ_{2, 3} \in (0, 1]$. Here $A_{p, \mathcal{R}}$ and $A_{p, \mathcal{Z}}$ respectively denote the class of of strong $A_p$ weights and the class of Zygmund $A_p$ weights. Beyond that, under similar bilinear assumptions, we prove bilinear Calderón-Zygmund operators associated with Zygmund dilations are compact from $L^{p_1}(\mathbb{R}^3) \times L^{p_2}(\mathbb{R}^3)$ to $L^p(\mathbb{R}^3)$ for all $p_1, p_2 \in (1, \infty)$, where $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}$. The core of the proof is a compact dyadic representation, which asserts that under the hypotheses above, a (bilinear) Calderón-Zygmund operator associated with Zygmund dilations can be represented an average of some compact (bilinear) dyadic shifts of Zygmund nature. This further deepens our understanding of the compactness of singular integral operators.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 3.1
• Definition 3.2