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

TL;DR

This paper characterizes pairs of weights on the real line and upper half-plane for which the Cauchy transform is bounded between weighted L2 spaces, extending recent two-weight inequality results and applying to model space embeddings and composition operators.

Contribution

It provides a new characterization of two-weight inequalities for the Cauchy transform, generalizing previous results for the Hilbert transform and applying to model space and composition operator analysis.

Findings

Characterization of weight pairs satisfying the $A_2$ condition.

Extension of two-weight inequalities to the Cauchy transform.

Applications to model space embeddings and composition operators.

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.