Converse to bounded Cauchy transform for Clark measures

Determine whether the following converse holds: for an inner function u on the unit disk with Clark measure σ, if the truncated Cauchy transform C_σ defined on L^2(σ) by C_σ f(ζ) = ∫_{𝕋\{ζ}} f(s)/(1 − s̄ζ) dσ(s) is bounded on L^2(σ), then u must be a one-component inner function.

Background

The paper proves that if u is a one-component inner function, then the truncated Cauchy transform C_σ associated to its Clark measure σ is bounded on L2(σ) (Theorem 4). This relies on Bessonov’s characterization of one-component inner functions, which includes a T(1)-type condition, and on Tolsa’s criterion for the boundedness of the Cauchy transform.

The authors note that the relationship between inner functions and the boundedness of C_σ is not fully understood and pose the converse direction as an explicit open problem: does boundedness of C_σ characterize the one-component property of u? A positive answer would provide a new operator-theoretic characterization of one-component inner functions.

References

In this regard, we conclude this section with an open problem. Let $\sigma$ be the Clark measure associated with $u$. Is it true that if $\mathcal{C}_\sigma$ is bounded on $L2(\sigma)$, then $u$ is a one-component inner function?

— Infinitely supported harmonically weighted Dirichlet spaces which are de Branges Rovnyak spaces  (2509.04907 - Bellavita et al., 5 Sep 2025) in End of Section 4 (Proof of Theorem 4; Theorem \ref{T:main2})