Off-diagonal estimates for bi-parameter commutators

Abstract

We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces
 \([b,T]\colon L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}\)
 in the off-diagonal situation \(q_i,p_i\in(1,\infty)\) where we allow \(q_i\not= p_i\). Boundedness is fully characterized in terms of oscillatory testing conditions on the function \(b\) for a total seven out of the nine possible arrangements of the integrability exponents.