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Une généralisation du théorème de Calderón sur l’intégrale de Cauchy. (French) Zbl 0541.42008
Fourier analysis, Proc. Semin., El Escorial 1979, 87-116 (1980).
Let \(\mathcal G\) be the group of increasing homeomorphisms \(h\) of \(\mathbb{R}\) that preserve the class of sets of measure zero and operate on BMO \((f\mathbb{O}h\in \mathrm{BMO}\) for \(f\in\mathrm{BMO}\). The map \(h\mapsto \log h'\) (derivative in the sense of distributions) takes \(\mathcal G\) onto an open set \(U\) in \(\mathrm{BMO}_{\mathbb{R}}\).
Let \(H\) denote the Hilbert transform and, for \(h\in\mathcal G\), \(V_h\) the operator \(f\mapsto f{\mathbb{O}}h\) on BMO.
Theorem: The map \(\Lambda: \log h' \mapsto V_h\) from \(U\) to \(\mathcal L(\mathrm{BMO})\) is real analytic.
This theorem is applied to give a direct sum decomposition of \(\mathrm{BMO}(\Gamma)\) \((\Gamma\) a plane curve subject to a so called chord-arc condition) and to prove boundedness of singular Calderón integral operators.
[For the entire collection see Zbl 0451.00015.]

42A50 Conjugate functions, conjugate series, singular integrals