A compact $T1$ theorem for Calder\'{o}n-Zygmund operators associated with Zygmund dilations

TL;DR

This paper establishes a compact version of the T1 theorem for Zygmund-type Calderón-Zygmund operators, demonstrating their compactness on weighted L^p spaces and extending results to bilinear operators, using a dyadic representation approach.

Contribution

It introduces a compact T1 theorem for Zygmund Calderón-Zygmund operators and proves their compactness on weighted L^p spaces, including bilinear cases, via a dyadic representation method.

Findings

Proved compactness of Zygmund Calderón-Zygmund operators on weighted L^p spaces.

Extended compactness results to bilinear Calderón-Zygmund operators.

Developed a compact dyadic representation for these operators.

Abstract

We develop a compact version of $T1$ theorem for singular integrals of Zygmund type on $\mathbb{R}^3$. More specifically, if a $(D_{\theta}, \delta_1, \delta_{2, 3})$-Calder\'{o}n-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 $\theta, \delta_1, \delta_{2, 3} \in (0, 1]$, or (ii) $p \in (1, \infty)$, $w \in A_{p, \mathcal{Z}}$, $\theta = \delta_1 = 1$, and $\delta_{2, 3} \in (0, 1]$. 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 Calder\'{o}n-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) Calder\'{o}n-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.