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

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.