A characterization of lines among Lipschitz graphs. (English) Zbl 0841.42009

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.


42B25 Maximal functions, Littlewood-Paley theory
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane


Zbl 0788.42007
Full Text: DOI EuDML


[1] Chevalier, L.: Espace BMO, mesures de Carleson, fonctionsg de Littlewood-Paley g?n?ralis?es et conditions d’annulation. Math. Ann.297 (1993), 269-288 · Zbl 0788.42007
[2] David, G., Semmes S.:Analysis of and on Uniformly Rectifiable Sets, Providence R.I., A.M.S. Mathematical surveys and monographs38 (1993) · Zbl 0832.42008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.