Geodesic
On the Nikol'skii Classes of Polyharmonic Functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Investigations in the theory of differentiable functions of many variables and its applications. Part 18, Tome 227 (1999), pp. 43-55.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper is devoted to the study of the properties of polyharmonic functions defined on the unit ball $D^m$ of the Euclidean space $\mathbb R^m$, $D^m = \{x\in\mathbb R^m\mid |x|1\}$. With the help of the well-known Almansi decomposition, the polyharmonic function is represented as a sum of components, each of which has a simple form. The main idea, developed in [1–3], is that, under a suitable choice of components, the behavior of these components near the boundary of the ball $D^m$ is no worse than that of the polyharmonic function itself. Here, in view of the smoothness at internal points, the boundary behavior of a polyharmonic function is naturally characterized by its membership in a certain functional class on $D^m$. 
@article{TM_1999_227_a1,
     author = {K. O. Besov},
     title = {On the {Nikol'skii} {Classes} of {Polyharmonic} {Functions}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {43--55},
     publisher = {mathdoc},
     volume = {227},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_1999_227_a1/}
}
TY  - JOUR
AU  - K. O. Besov
TI  - On the Nikol'skii Classes of Polyharmonic Functions
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 1999
SP  - 43
EP  - 55
VL  - 227
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_1999_227_a1/
LA  - ru
ID  - TM_1999_227_a1
ER  - 
%0 Journal Article
%A K. O. Besov
%T On the Nikol'skii Classes of Polyharmonic Functions
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 1999
%P 43-55
%V 227
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_1999_227_a1/
%G ru
%F TM_1999_227_a1
K. O. Besov. On the Nikol'skii Classes of Polyharmonic Functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Investigations in the theory of differentiable functions of many variables and its applications. Part 18, Tome 227 (1999), pp. 43-55. http://geodesic.mathdoc.fr/item/TM_1999_227_a1/

[1] Dolzhenko E. P., “O granichnom povedenii komponent polianaliticheskoi funktsii”, Dokl. RAN, 338:5 (1994), 585–588 | MR | Zbl

[2] Dolzhenko E. P., “O granichnom povedenii komponent polianaliticheskoi funktsii”, Mat. zametki, 63:6 (1998), 821–834 | MR | Zbl

[3] Besov K. O., “O granichnom povedenii komponent poligarmonicheskikh funktsii”, Mat. zametki, 64:4 (1998), 518–530 | MR | Zbl

[4] Dolzhenko E. P., Danchenko V. I., “O granichnom povedenii reshenii obobschennogo uravneniya Koshi–Rimana”, Vestn. MGU. Matematika. Mekhanika, 1998, no. 3, 16–24 | MR

[5] Besov K. O., “O nalichii interferentsii mezhdu komponentami bigarmonicheskikh funktsii”, Vestn. MGU. Matematika. Mekhanika, 1999, no. 6 | MR

[6] Mikhailov V. P., “O suschestvovanii predelnykh znachenii na granitse oblasti reshenii poligarmonicheskogo uravneniya”, Mat. sb., 187:11 (1996), 89–114 | MR | Zbl

[7] Almansi E., “Sull'integrazione dell'equazione differenziale $\Delta ^n=0$”, Ann. mat., ser. 3, 2 (1899), 1–59

[8] Nicolesco M., Les fonctions polyharmoniques, Hermann, Paris, 1936

[9] Muramatu T., “On Besov spaces and Sobolev spaces of generalized functions defined on general region”, Publ. Res. Inst. Math. Sci. Kyoto Univ., 9:2 (1974), 325–396 | DOI | MR | Zbl

[10] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[11] Bitsadze A. V., “O nekotorykh svoistvakh poligarmonicheskikh funktsii”, Dif. uravneniya, 24:5 (1988), 825–831 | MR | Zbl

[12] Bergman S., Shiffer M., Kernel functions and elliptic differential equations in mathematical physics, Acad. Press, N. Y., 1953 | MR | Zbl