Hardy spaces for the conjugated Beltrami equation in a doubly connected domain

Messoud Efendiev (1) , Emmanuel Russ (2)

1 IBB - Institut für Biomathematik und Biometrie
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

We consider Hardy spaces associated to the conjugated Beltrami equation on doubly connected planar domains. There are two main differences with previous studies. First, while the simple connectivity plays an important role in the simply connected case, the multiple connectivity of the domain leads to unexpected difficulties. In particular, we make strong use of a suitable parametrization of an analytic function in a ring by its real part on one part of the boundary and by its imaginary part on the other. Then, we allow the coefficient in the conjugated Beltrami equation to belong to $W^{1,q}$ for some $q\in (2,+\infty]$, while it was supposed to be Lipschitz in the simply connected case. We define Hardy spaces associated with the conjugated Beltrami equation and solve the corresponding Dirichlet problem. The same problems for generalized analytic function are also solved.

Keywords: Hardy spaces; doubly connected domain; Dirichlet problem; analytic projection

Domains: Analysis of PDEs [math.AP]; Complex Variables [math.CV]

Messoud Efendiev, Emmanuel Russ. Hardy spaces for the conjugated Beltrami equation in a doubly connected domain. Journal of Mathematical Analysis and Applications, 2011, 383 (2), pp.439-450. ⟨10.1016/j.jmaa.2011.05.031⟩. ⟨hal-00605932⟩