Geodesic
On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple Carathéodory domains
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 47, Tome 480 (2019), pp. 62-72 
E. V. Borovik; K. Yu. Fedorovskiy. On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple Carathéodory domains. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 47, Tome 480 (2019), pp. 62-72. http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a3/
@article{ZNSL_2019_480_a3,
     author = {E. V. Borovik and K. Yu. Fedorovskiy},
     title = {On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple {Carath\'eodory} domains},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {62--72},
     year = {2019},
     volume = {480},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a3/}
}
TY  - JOUR
AU  - E. V. Borovik
AU  - K. Yu. Fedorovskiy
TI  - On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple Carathéodory domains
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 62
EP  - 72
VL  - 480
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a3/
LA  - ru
ID  - ZNSL_2019_480_a3
ER  - 
%0 Journal Article
%A E. V. Borovik
%A K. Yu. Fedorovskiy
%T On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple Carathéodory domains
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 62-72
%V 480
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_480_a3/
%G ru
%F ZNSL_2019_480_a3

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Carathéodory domains. This operator is viewed as one acting from a space of functions of Lipschitz–Hölder type, defined by a general modulus of continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question, which are also obtained in the paper; they generalize and refine the results of the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019).

[1] P. V. Paramonov, “O $\mathrm{Lip}^m$- i $C^m$-otrazhenii garmonicheskikh funktsii otnositelno granits oblastei Karateodori v $\mathbb R^2$”, Vestnik MGTU im. N. E. Baumana. Ser. Estestvennye nauki, 2018, no. 4, 36–45 | MR

[2] K. Fedorovskiy, P. Paramonov, “On $\mathrm{Lip}^m$-reflection of harmonic functions over boundaries of simple Caratheodory domains”, Anal. Math. Phys., 2019 | DOI | MR

[3] H. Lebesgue, “Sur le problème de Dirichlet”, Rend. Circ. Mat. di Palermo, 29 (1907), 371–402 | DOI

[4] D. H. Armitage, “Reflection principles for harmonic and polyharmonic functions”, J. Math. Anal. Appl., 65 (1978), 44–55 | DOI | MR | Zbl

[5] D. Khavinson, H. S. Shapiro, “Remarks on the reflection principle for harmonic functions”, J. Anal. Math., 54 (1990), 60–76 | DOI | MR | Zbl

[6] D. Khavinson, “On reflection of harmonic functions in surfaces of revolution”, Complex Var. Theory Appl., 17 (1991), 7–14 | MR | Zbl

[7] P. Ebenfelt, D. Khavinson, “On point-to-point reflection of harmonic functions across real-analytic hypersurfaces in $\mathbb R^n$”, J. Anal. Math., 68 (1996), 145–182 | DOI | MR | Zbl

[8] S. J. Gardiner, H. Render, “A reflection result for harmonic functions which vanish on a cylindrical surface”, J. Math. Anal. Appl., 443:1 (2016), 81–91 | DOI | MR | Zbl

[9] P. V. Paramonov, “O $C^1$-prodolzhenii i $C^1$-otrazhenii subgarmonicheskikh funktsii s oblastei Lyapunova–Dini na $\mathbb R^N$”, Matem sb., 199:12 (2008), 79–116 | DOI | Zbl

[10] E. H. Johnston, “The boundary modulus of continuity of harmonic functions”, Pacific J. Math., 90:1 (1980), 87–98 | DOI | MR | Zbl

[11] L. A. Rubel, A. L. Shields, B. A. Taylor, “Mergelyan sets and the modulus of continuity of analytic functions”, J. Approx. Theor., 15 (1975), 23–40 | DOI | MR | Zbl

[12] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Second Edition, Springer-Verlag, New York, 2000 | MR

[13] A. Beurling, Études sur un Problème de Majoration, Almquist and Wiksell, Upsala, 1933 | Zbl