On the compactness of the bi-commutator

We prove compactness results and characterizations for the bi-commutator [T₁,[b, T₂]] of a symbol b and two non-degenerate Calder\'on-Zygmund singular integral operators T₁, T₂. Our strategy for proving sufficient conditions for compactness is to first establish them in the mixed-norm Lp₁Lp₂→ Lq₁Lq₂ off-diagonal case with pᵢ < qᵢ, and then extend these to other exponents, including the diagonal pᵢ = qᵢ, with a new extrapolation argument. In particular, the natural product VMO condition is obtained as a sufficient condition in the diagonal. A full characterization is obtained, both in terms of a vanishing mean oscillation type condition and in terms of the approximability of the symbol, whenever the inequality pᵢ ≤ qᵢ is strict for at least one index. The extrapolation scheme for proving sufficiency requires us to prove new approximation results in relevant bi-parameter function spaces that are of independent interest. The necessity results are obtained by carefully combining recent rectangular approximate weak factorization methods with a classical idea of Uchiyama.