“…The awf approach for off-diagonal estimates has been generalized to the bi-parameter setting in Airta, Hytönen, Li, Martikainen and Oikari [1] and Oikari [19]. In [1] and [19], respectfully, the commutators T 2 , [b, T 1 ] , b, T : L p 1 (R d 1 ; L p 2 (R d 2 )) → L q 1 (R d 1 ; L q 2 (R d 2 )), (1.6) where 1 < p 1 , p 2 , q 1 , q 2 < ∞, and T 1 , T 2 are non-degenerate one-parameter CZOs and T is a non-degenerate bi-parameter CZO, were treated. The adaptability of the awf method to the bi-parameter settings was not effortless and for both of the commutator on the line (1.6) the characterization of some cases is still open.…”