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We characterize those pairs of weights $ \sigma $ on $ \mathbb{R}$ and $ \tau$ on $ \mathbb{C}_+$ for which the Cauchy transform $\mathsf{C}_{\sigma} f (z)\equiv \int_{\mathbb{R}} \frac {f(x)} {x-z} \; \sigma (dx)$, $ z\in\mathbb{C}_+$, is bounded from $L ^2(\mathbb{R};\sigma)$ to $L^{2}(\mathbb{C}_+; \tau)$. The characterization is in terms of an $A_2$condition on the pair of weights and testing conditions for the transform,extending the recent solution of the two weight inequality for the Hilberttransform. As corollaries of this result we derive (1) a characterization ofembedding measures for the model space $K_\vartheta$, for arbitrary innerfunction $ \vartheta $, and (2) a characterization of the (essential) norm ofcomposition operators mapping $K_\vartheta$ into a general class of Hardy andBergman spaces.

Two Weight Inequalities for the Cauchy Transform from $ \mathbb{R}$ to $ \mathbb{C}_+$