Using Bellman function techniques, we obtain the optimal dependence
of the operator norms in $L^2(\mathbb{R})$ of the Haar multipliers $T_w^t$
on the corresponding $RH^d_2$ or $A^d_2$ characteristic of the
weight $w$, for $t=1,\pm 1/2$. These results can be viewed as
particular cases of estimates on homogeneous spaces $L^2(vd\sigma)$,
for $\sigma$ a doubling positive measure and $v\in A^d_2(d\sigma)$,
of the weighted dyadic square function $S_{\sigma}^d$. We show
that the operator norms of such square functions in $L^2(v d\sigma)$
are bounded by a linear function of the $A^d_2(d\sigma )$
characteristic of the weight $v$, where the constant depends only on
the doubling constant of the measure $\sigma$. We also show an
inverse estimate for $S_{\sigma}^d$. Both results are known when
$d\sigma=dx$. We deduce both estimates from an estimate for the Haar
multiplier $(T_v^{\sigma})^{1/2}$ on $L^2(d\sigma)$ when $v\in
A_2^d(d\sigma)$, which mirrors the estimate for $T_w^{1/2}$ in
$L^2(\mathbb{R})$ when $w\in A^d_2$. The estimate for the Haar multiplier
adapted to the $\sigma$ measure, $(T_v^{\sigma})^{1/2}$, is proved
using Bellman functions. These estimates are sharp in the sense
that the rates cannot be improved and be expected to hold for all
$\sigma$, since the particular case $d\sigma=dx$, $v=w$, correspond
to the estimates for the Haar multipliers $T^{1/2}_w$ proven to be
sharp.