Off-diagonal estimates for bilinear commutators

We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1<p,q<\infty$ and $r>\frac{1}{2},$ boundedness of the bilinear commutator $[b,T]_1(f,g) = bT(f,g) - T(bf,g).$ Our arguments cover almost all arrangements of the integrability exponents $p,q,r,$ with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.