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The conjugate function in plane curves. (English) Zbl 0615.46051
Let $$\Gamma =\partial \Omega$$ be a rectifiable Jordan curve and let $$\phi$$ be the normalized conformal mapping from the unit disc D onto $$\Omega$$. In this paper the conjugate function operator on $$\Gamma$$ is defined in a natural way and the following result is obtained: ”The curves such that log $$| \phi '|$$ belongs to the closure of $$L^{\infty}$$ in BMO are exactly those for which the boundedness of the conjugate function operator is equivalent to the fact that $$w\in A_ p(\Gamma)''$$. The quasiregular curves are examples of such curves.
##### MSC:
 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30C20 Conformal mappings of special domains
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