Jordan domains with a rectifiable arc in their boundary
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Jordan domains with a rectifiable arc in their boundary Vasiliki Liontou1
· Vassili Nestorodis2
Received: 21 January 2019 / Accepted: 8 July 2019 © Fondation Carl-Herz and Springer Nature Switzerland AG 2019
Abstract We show that if an open arc J of the boundary of a Jordan domain Ω is rectifiable, then the derivative Φ of the Riemann map Φ : D → Ω from the open unit disk D onto Ω behaves as an H 1 function when we approach the arc Φ −1 (J ), where J is any compact subarc of J . Keywords Extendability · Total unboundedness · Generic property · Function space · Localization · Riemann map · rectifiable curve · Jordan domain · Hardy class H 1 · reflection principle Mathematics Subject Classification 30H10 Résumé Nous démontrons que si un arc ouvert J de la frontière d’ un domaine de Jordan Ω est rectifiable, alors la dérivée Φ de la fonction de Riemann Φ entre le disque unité ouvert D sur Ω se comporte comme une fonction de classe de Hardy H 1 , quand on approche le sous-ensemble Φ −1 (J ) où J est un sous-ensemble compact de J .
1 Introduction In [7] the Reflection principle has been used in order to prove that if a conformal collar, bounded by a Jordan arc δ has some nice properties, then any other conformal collar of δ on the same side has the same nice properties. We use the same method in order to generalize a well-known theorem about rectifiable Jordan curves, [3].
To the memory of Professor Alain Dufresnoy.
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Vasiliki Liontou [email protected] Vassili Nestorodis [email protected]
1
Department of Mathematics, University of Toronto, 40 St George st., Toronto, ON M5S 2E4, Canada
2
Department of Mathematics, National and Kapodistrian University of Athens, Panepistemiopolis, 157 84 Athens, Greece
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V. Liontou, V. Nestorodis
Theorem 1.1 Let τ be a Jordan curve and Φ : D → Ω be a Riemann map from the open unit disc D onto the interior Ω of τ . Then 1 and 2 below are equivalent: 1. τ is rectifiable. 2. The derivative Φ belongs to the Hardy class H 1 . The generalization we obtain is that if τ is not rectifiable, but an open arc J of it has finite length, then the derivative Φ behaves as an H 1 function when we approach the compact subsets of the arc Φ −1 (J ) ⊂ {z ∈ C : |z| = 1}. In the proof we combine the statement of Theorem 1.1 with the Reflection principle, [1]. The above suggests that the Hardy spaces H p on the disc can be generalized to larger spaces containing exactly all holomorphic functions f on the open unit disc D, such that b sup | f (r eit )| p dt < +∞, for some fixed a, b with a < b < a + 2π. One can inves0 0. Thus, t1 ∈ [t1 − δ, t1 + δ] and [z 1 , γ (t1 )] ∩ γ (I ) = {γ (t1 )}. Therefore, there exists an open segment inside the interior of γ , which joins z 1 with γ (t1 ). We repeat the procedure for b < t2 − δ < t2 + δ < B and will find γ (t2 ) and z 2 in the interior of G of γ , such that the open segment (z 2 , γ (t2 )) is included in G. Therefore, there exists a polygonal line W that connects z 2 and z 1 in G. It can easily be proven that t
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