A characterization of compactness via bilinear $T1$ theorem

Abstract

In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calderón-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear Calderón-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 $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}>0$ 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 Calderón-Zygmund kernel, (H2) $T$ satisfies the weak compactness property, and (H3) $T(1,1),T^{\ast 1}(1,1),T^{\ast 2}(1,1)\in \mathrm{CMO}({\mathbb{R}}^n)$. This is also equivalent to the endpoint compactness: (1) $T$ is compact from $L^1(w_1)\times L^1(w_2)$ to $L^{\frac{1}{2},\infty }(w^{\frac{1}{2}})$ 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 Calderón-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.