Analytic balayage of measures, Carathéodory domains, and badly approximable functions in Lp

On Dirichlet problem for second-order elliptic equations in the plane and uniform approximation problems for solutions of such equations

“…It was shown that the class of Nevanlinna domains is rather big in spite of the fact that the definition of a Nevanlinna domain imposes quite rigid condition to the boundary of the domain under consideration. For instance, there exists such Nevanlinna domains G that the Hausdorff dimension of ∂G could take any value in [1,2] (see [5], Theorem 3).…”

“…For certain non-Carathéodory compact sets of a special form the sufficient approximability conditions were obtained in [7], [11] and [10]. One interesting and helpful tool using in these works is the concept of an analytic balayage of measures which was introduced by D. Khavinson [16], which was rediscovered in [11] in a slightly different terms, and which was studied by several authors both as an object of an independent interest and in connection with properties of badly approximable functions in L p (T) (see [1] and bibliography therein). It seems to us interesting and appropriate to note this point.…”

“…Next we do the same thing for the domain G τ := T G, which is a Carathéodory one, and for which it holds ∂ a G τ = T (∂ a G). Let ϕ τ be some conformal mapping from D onto G τ such that 1 ∈ F(ϕ τ ) and T ϕ(1) = ϕ τ (1). As previously we denote by ψ τ the inverse mapping for ϕ τ in G τ and by ω τ the measure ϕ τ (dξ) on ∂G τ .…”