A note on the off-diagonal Muckenhoupt-Wheeden conjecture

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderon-Zygmund operators. Namely, given 1 < p < q < ∞ and a pair of weights (u, v), if the Hardy-Littlewood maximal function satisfies the following two weight inequalities: M : L(v) → L(u) and M : L ′ (u ′ ) → L ′ (v ′ ), then any Calderon-Zygmund operator T and its associated truncated maximal operator T? are bounded from L (v) to L(u). Additionally, assuming only the second estimate for M then T and T? map continuously L(v) into Lq,∞(u). We also consider the case of generalized Haar shift operators and show that their off-diagonal two weight estimates are governed by the corresponding estimates for the dyadic Hardy-Littlewood maximal function.