Off-diagonal estimates for commutators of bi-parameter singular integrals

We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces $$[b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$$ in the off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$ Boundedness is fully characterized for several arrangements of the integrability exponents with some open problems presented.