On Lip^m - and C^m -reflection of harmonic functions

It is planned to discuss one problem about reflection of harmonic functions over
boundaries of simple Carathéodory domains in R^n , n > 2. A bounded domain G ⊂ R^n is
said to be a simple Carathéodory domain, if the following three conditions are fulfilled: (i)
the set Ω = R^n \ G is connected; (ii)∂G = ∂Ω; and (iii) if n > 3, then both domains G and
Ω are regular with respect to the classical Dirichlet problem for harmonic functions (in the
case n = 2 the latter condition holds for every domain G satisfying (ii)). Given a function
f harmonic in G and continuous on G, the solution g of the Dirichlet problem in Ω with
the boundary condition g| ∂Ω = f | ∂G is called a reflection of f over ∂G. The operator R_G
that maps f to g is called the harmonic reflection operator associated with G.
We are interested in conditions on G that yield Lip^m - and C^m -continuity of the operator
R_D for m ∈ (0, 1). The roots of these questions are traced to the works by M. Melnikov,
P. Prarmonov and J. Verdera about C^m -extension of harmonic and subharmonic functions
from compact sets in R^n with norm preservation. The obtained results are based on new
criteria of Lip^m - and C^m -continuity of the Poisson operator in domains under consideration
which will be also presented and discussed in the talk. Moreover, it is planned to present
new sufficient conditions for C^m -approximability of functions by harmonic polynomials on
boundaries of Carathéodory domains.

The talk is base in join works with P. Paramonov.