A note on the off-diagonal Muckenhoupt-Wheeden conjecture

Abstract

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\'on-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: $$ M : L^p(v) \rightarrow L^q(u) \quad \text{and} \quad M: L^{q'}(u^{1-q'}) \rightarrow L^{p'}(v^{1-p'}), $$ then any Calder\'on-Zygmund operator $T$ and its associated truncated maximal operator $T_\star$ are bounded from $L^p(v)$ to $L^q(u)$. Additionally, assuming only the second estimate for $M$ then $T$ and $T_\star$ map continuously $L^p(v)$ into $L^{q,\infty}(u)$. 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.