A characterization of compactness via bilinear $T1$ theorem

TL;DR

This paper characterizes the weighted compactness of bilinear Calderón-Zygmund operators using a new bilinear T1 theorem, involving kernel compactness, weak compactness, and BMO/CMO conditions, with applications to various bilinear operators.

Contribution

It provides a complete characterization of weighted compactness for bilinear Calderón-Zygmund operators via a bilinear T1 theorem, including new representation and extrapolation techniques.

Findings

Characterization of compactness via kernel and BMO/CMO conditions.

Representation of operators as averages of dyadic shifts and paraproducts.

Applicability demonstrated through examples like bilinear paraproducts and pseudo-differential operators.

Abstract

In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear Calder\'{o}n-Zygmund kernel. We prove that $T$ can be extended to a compact bilinear operator from $L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2})$ to $L^p(w^p)$ for all exponents $\frac1p = \frac{1}{p_1} + \frac{1}{p_2}>0$ with $p_1, p_2 \in (1, \infty]$ and for all weights $(w_1, w_2) \in A_{(p_1, p_2)}$ if and only if the following hypotheses hold: (H1) $T$ is associated with a compact bilinear Calder\'{o}n-Zygmund kernel, (H2) $T$ satisfies the weak compactness property, and (H3) $T(1,1), T^{*1}(1,1), T^{*2}(1,1) \in \mathrm{CMO}(\mathbb{R}^n)$. This is also equivalent to the endpoint compactness: (1) $T$ is compact from $L^1(w_1) \times L^1(w_2)$ to $L^{\frac12, \infty}(w^{\frac12})$ for all $(w_1, w_2) \in A_{(1, 1)}$, or (2) $T$ is compact from $L^{\infty}(w_1^{\infty}) \times L^{\infty}(w_2^{\infty})$ to $\mathrm{CMO}_{\lambda}(w^{\infty})$ for all $(w_1, w_2) \in A_{(\infty, \infty)}$. Besides, any of these properties is equivalent to the fact that $T$ admits a compact bilinear dyadic representation. Our main approaches consist of the following new ingredients: (i) a resulting representation of a compact bilinear Calder\'{o}n-Zygmund operator as an average of some compact bilinear dyadic shifts and paraproducts; (ii) extrapolation of endpoint compactness for bilinear operators; and (iii) compactness criterion in weighted Lorentz spaces. Finally, to illustrate the applicability of our result, we demonstrate the hypotheses (H1)-(H3) through examples including bilinear continuous/dyadic paraproducts, bilinear pseudo-differential operators, and bilinear commutators.