Let
$\sigma $ and
$\omega $ be positive Borel
measures on
$\mathbb{R}$
with
$\sigma $ doubling.
Suppose first that
$1<p\le 2$.
We characterize boundedness of certain maximal truncations of the Hilbert transform
${T}_{\u266e}$ from
${L}^{p}\left(\sigma \right)$ to
${L}^{p}\left(\omega \right)$ in terms of the
strengthened
${A}_{p}$
condition
$${\left({\int}_{\mathbb{R}}{s}_{Q}{\left(x\right)}^{p}d\omega \left(x\right)\right)}^{1\u2215p}{\left({\int}_{\mathbb{R}}{s}_{Q}{\left(x\right)}^{{p}^{\prime}}d\sigma \left(x\right)\right)}^{1\u2215{p}^{\prime}}\le C\leftQ\right,$$
where
${s}_{Q}\left(x\right)=\leftQ\right\u2215\left(\leftQ\right+x{x}_{Q}\right)$,
and two testing conditions. The first applies to a restricted class of functions and is a
strongtype testing condition,
$${\int}_{Q}{T}_{\u266e}\left({\chi}_{E}\sigma \right){\left(x\right)}^{p}d\omega \left(x\right)\le {C}_{1}{\int}_{Q}d\sigma \left(x\right)\phantom{\rule{1em}{0ex}}\text{forall}E\subset Q,$$
and the second is a weaktype or dual interval testing condition,
$${\int}_{Q}{T}_{\u266e}\left({\chi}_{Q}f\sigma \right)\left(x\right)d\omega \left(x\right)\le {C}_{2}{\left({\int}_{Q}f\left(x\right){}^{p}d\sigma \left(x\right)\right)}^{1\u2215p}{\left({\int}_{Q}d\omega \left(x\right)\right)}^{1\u2215{p}^{\prime}}$$
for all intervals
$Q$
in
$\mathbb{R}$ 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
$${\int}_{\mathbb{R}}{\left(\sum _{r=1}^{\infty}\left{I}_{r}{}_{\sigma}\right{I}_{r}{}^{{p}^{\prime}1}\sum _{\ell =0}^{\infty}\frac{{2}^{\ell}}{{\left({I}_{r}\right)}^{\left(\ell \right)}}{\chi}_{{\left({I}_{r}\right)}^{\left(\ell \right)}}\left(y\right)\right)}^{p}d\omega \left(y\right)\le C\sum _{r=1}^{\infty}\left{I}_{r}{}_{\sigma}\right{I}_{r}{}^{{p}^{\prime}},$$
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
${\mathbb{R}}^{n}$
when
$\sigma $ is
doubling and
$1<p<\infty $.
Finally, we characterize the weaktype two weight inequality for certain maximal singular
integrals
${T}_{\u266e}$
in
${\mathbb{R}}^{n}$ when
$1<p<\infty $, without the doubling
assumption on
$\sigma $,
in terms of analogues of the second testing condition and the
${A}_{p}$
condition.
