A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

Abstract

Let $\sigma$ and $\omega$ be positive Borel measures on $ℝ$ with $\sigma$ doubling. Suppose first that $1<p\le 2$. We characterize boundedness of certain maximal truncations of the Hilbert transform $T_{♮}$ from $L^p\left(\sigma \right)$ to $L^p\left(\omega \right)$ in terms of the strengthened $A_p$ condition

$${\left({\int }_{ℝ}{s}_Q{\left(x\right)}^{p}d\omega \left(x\right)\right)}^{1∕p}{\left({\int }_{ℝ}{s}_Q{\left(x\right)}^{p^{\prime }}d\sigma \left(x\right)\right)}^{1∕p^{\prime }}\le C\left|Q\right|,$$

where $s_Q\left(x\right)=\left|Q\right|∕\left(\left|Q\right|+|x-x_Q|\right)$, and two testing conditions. The first applies to a restricted class of functions and is a strong-type testing condition,

$${\int }_Q{T}_{♮}\left({\chi }_E\sigma \right){\left(x\right)}^{p}d\omega \left(x\right)\le C_1{\int }_{Q}d\sigma \left(x\right)\text{ for all }E\subset Q,$$

and the second is a weak-type or dual interval testing condition,

$${\int }_Q{T}_{♮}\left({\chi }_{Q}f\sigma \right)\left(x\right)d\omega \left(x\right)\le C_2{\left({\int }_Q|f\left(x\right){|}^{p}d\sigma \left(x\right)\right)}^{1∕p}{\left({\int }_{Q}d\omega \left(x\right)\right)}^{1∕p^{\prime }}$$

for all intervals $Q$ in $ℝ$ and all functions $f\in L^p\left(\sigma \right)$. In the case $p>2$ the same result holds if we include an additional necessary condition, the Poisson condition

$${\int }_{ℝ}{\left(\sum _{r=1}^{\infty }\left|I_r{|}_{\sigma }\right|I_r{|}^{p^{\prime }-1}\sum _{\ell =0}^{\infty }\frac{2^{-\ell }}{|{\left(I_r\right)}^{\left(\ell \right)}}|{\chi }_{{\left(I_r\right)}^{\left(\ell \right)}}\left(y\right)\right)}^{p}d\omega \left(y\right)\le C\sum _{r=1}^{\infty }\left|I_r{|}_{\sigma }\right|I_r{|}^{p^{\prime }},$$

for all pairwise disjoint decompositions $Q={\bigcup }_{r=1}^{\infty }{I}_r$ of the dyadic interval $Q$ into dyadic intervals $I_r$. We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in ${ℝ}^n$ when $\sigma$ is doubling and $1<p<\infty$. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals $T_{♮}$ in ${ℝ}^n$ when $1<p<\infty$, without the doubling assumption on $\sigma$, in terms of analogues of the second testing condition and the $A_p$ condition.