The conjugate function in plane curves. (English) Zbl 0615.46051

Guadalupe, José J.; Rezola, M. Luisa

Can. Math. Bull. 31, No. 2, 147-152 (1988).

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

Keywords:

rectifiable Jordan curve; normalized conformal mapping; BMO; boundedness of the conjugate function operator; quasiregular curves