On Lip^m - and C^m -reflection of harmonic functions (Konstantin Fedorovskiy - Bauman Moscow State Technical University, Saint-Petersburg State University)

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.