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.