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

Abstract

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 Hilbert transform. As corollaries of this result we derive (1) a characterization of embedding measures for the model space $K_\vartheta$, for arbitrary inner function $ \vartheta $, and (2) a characterization of the (essential) norm of composition operators mapping $K_\vartheta$ into a general class of Hardy and Bergman spaces.