On Lip^m- and C^m-Reflection of Harmonic Functions with Respect to Boundaries of Caratheodory Domains in R²

In this paper a number of sharp necessary and sufficient conditions for Lip^m- and C^m-continuity of operators of harmonic reflection of functions over boundaries of simple Caratheodory domains in R² are obtained. Let us state the main result of this paper in a simplified form. For an arbitrary real function f that is harmonic in a Jordan domain and continuous in its closure, let R(f) be the solution of the Dirichlet problem in the domain Ω with the boundary function f|_∂Ω, where ∂Ω is the boundary of Ω. The function R(f) is said to be a harmonic reflection of the function f with respect to the boundary ∂D = ∂Ω of the domain D, and the operator R: f — R(f) is said to be a harmonic reflection operator. Let now D has a piecewise smooth boundary, and suppose πα ∈ (0,π] to be the value of the minimal inner angle of the domain D (it means that πα is the minimal value among all inner angles V_a, a∈∂D, formed by couples of distinct rays tangent to ∂D with vertexes at a). Let m_α = 1/(2-α). Then for all (m,m’) such that 0 < m’ < m_α or 0 < m’ < m_α ≤ m ≤ 1 the operator R is (m, m’)-continuous, and it is not the case for m_α < m’ ≤ m ≤ 1. The (m, m’)-continuity property means that for any function f harmonic in D we have R(f) ∈ Lip^m (Ω). Moreover, the Lip^m (Ω)-norm of R(f) has to be estimated (up to certain multiplicative constant) by the Lip^m (D)-norm of f

The work was carried out with the state financial support of the Ministry of Education and Science of the Russian Federation (project no. 1.3843.2017/4.6)

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