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 L^p₁L^p₂ L^q₁L^q₂ 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.