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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.]

##### MSC:
 42A50 Conjugate functions, conjugate series, singular integrals