A note on the off-diagonal Muckenhoupt-Wheeden conjecture
Abstract
We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calderón-Zygmund operators. Namely, given $1 < p < q < \infty$ and a pair of weights $(u; v)$, if the Hardy-Littlewood maximal function satisfies the following two
weight inequalities:
$$
M:L^p(v) \to L^q(u) \quad \mbox{and} \quad M: L^{q'} (u^{1-q'}) \to (v^{1-p'} );
$$
then any Calderón-Zygmund operator $T$ and its associated truncated maximal
operator $T_{*}$ are bounded from $M:L^p(v)$ to $L^q(u)$. Additionally, assuming only the
second estimate for $M$ then $T$ and $T_{*}$ map continuously $M:L^p(v)$ to $L^{q,\infty}(u)$
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.