A characterization of lines among Lipschitz graphs (Q1910173)

Let \(A\) be a Lipschitz function from \(\mathbb{R}\) into \(\mathbb{R}\). A study of the Littlewood-Paley \(g\)-function associated with the Cauchy kernel of a Lipschitz graph led the first author to conjecture [in Math. Ann. 297, No. 2, 269-288 (1993; Zbl 0788.42007)] the following result, that we prove in this paper: Theorem 1. Assume that, for every \(t> 0\), \[ \int^{+ \infty}_{- \infty} {dy\over (y+ i(A(y)- t))^2}= 0. \] Then \(A\) is an affine function. This result allows us to complete the proof of the following theorem: Theorem 2. Let \(r\) be the Cauchy kernel associated with a Lipschitz graph \(\Gamma\). Then the Littlewood-Paley operator \(g^2_r\) maps \(^\infty\) into BMO, and it maps BMO into BMO if and only if \(\Gamma\) is a line.