A characterization of compactness via bilinear T1 theorem

We establish a bilinear T1 theorem to characterize the weighted compactness of bilinear Calder\'on--Zygmund operators. Let T be a bilinear operator associated with a standard bilinear Calder\'on--Zygmund kernel. We demonstrate that T can be extended to a compact bilinear operator from L^p₁ (w₁^p₁) L^p₂ (w₂^p₂) to Lᵖ (wᵖ) for all exponents 1p = 1p₁ + 1p₂ with 1<p₁, p₂< and for all weights (w₁, w₂) A (₏䃑, ₏䃒) if and only if the following conditions hold: (i) T is associated with a compact bilinear Calder\'on--Zygmund kernel, (ii) T satisfies the weak compactness property, and (iii) T (1, 1), T^*1 (1, 1), T^*2 (1, 1) CMO (Rⁿ).