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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.

MSC:

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

Citations:

Zbl 0788.42007
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References:

[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
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