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

Abstract

International audienceWe 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

Dirichlet/Neumann problems and Hardy classes for the planar conductivity equation

“…Below we enumerate some properties that G p α (Ω) and H p ν (Ω) inherit from H p (Ω) via Proposition 2. These generalize results stressed in [11,27] for the simply or doubly connected case (except the last two which are not mentioned in [27]). Recall f defined on Ω has non tangential ("n.t.")…”

“…We conclude this section with a parameterization of G p α (Ω) by H p (Ω)-functions which proceeds differently from Proposition 2, and is fundamental to our approach of the Dirichlet problem. It was essentially obtained on the disk in [11] when σ ∈ W 1,∞ (Ω), and then carried over to the annulus in [27] under the assumption that r > 2 and p > r/(r − 2). It features the operator T α , defined for h ∈ L p (Ω) by the formula…”

Pseudo-holomorphic functions at the critical exponent

“…It is in this form that condition Ap is necessary and sufficient for weighted L p boundedness of the singular Cauchy integral operator on Γ, see [8,Chapter 5]. 14 , where C = C(Ω). Then h := ℓϕf lies in W 1,2 0 (Ω 1 ) and satisfies ∂h 2…”

“…Hardy classes for such functions were introduced in [35] and subsequently considered in [27,28,29,5] in the range of exponents 1 < p < ∞, see [14,30,16,4] for further generalizations to multiply connected domains. The connection between pseudo-holomorphic functions and conjugate Beltrami equations makes pseudo-holomorphic Hardy classes a convenient framework to solve Dirichlet problems with L p boundary data for isotropic conductivity equations [5,14,4]. These are also instrumental in [17,18,19,16] to approach certain inverse boundary problems.…”

“…Lemma 8.2 carry over mechanically to this more general setting, but the significance of condition A p with respect to the weighted L p continuity of the conjugate operator is no longer the same if Γ is nonsmooth14 . Such considerations are not needed in this paper, but we make use at some point of the following estimate showing that W…”