Off-diagonal estimates for bi-parameter commutators

Abstract

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 $q_i,p_i\in (1,\mathrm{\infty })$ where we allow $q_i\ne p_i$. 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.