Geodesic
The Traces of Bessel Potentials on Regular Subsets of Carnot Groups
Matematičeskie trudy, Tome 10 (2007) no. 2, pp. 19-61.

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

We prove the direct theorem on the traces of the Bessel potentials $L^\alpha_p$ defined on a Carnot group, on the regular closed subsets called Ahlfors $d$-sets. The result is convertible for integer $\alpha$, i.e., for the Sobolev spaces $W^\alpha_p$ (the converse trace theorem was proven in [1]). This theorem generalizes A. Johnsson and H. Wallin's results [2] for Sobolev functions and Bessel potentials on the Euclidean space. 
@article{MT_2007_10_2_a1,
     author = {S. K. Vodop'yanov and I. M. Pupyshev},
     title = {The {Traces} of {Bessel} {Potentials} on {Regular} {Subsets} of {Carnot} {Groups}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {19--61},
     publisher = {mathdoc},
     volume = {10},
     number = {2},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2007_10_2_a1/}
}
TY  - JOUR
AU  - S. K. Vodop'yanov
AU  - I. M. Pupyshev
TI  - The Traces of Bessel Potentials on Regular Subsets of Carnot Groups
JO  - Matematičeskie trudy
PY  - 2007
SP  - 19
EP  - 61
VL  - 10
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2007_10_2_a1/
LA  - ru
ID  - MT_2007_10_2_a1
ER  - 
%0 Journal Article
%A S. K. Vodop'yanov
%A I. M. Pupyshev
%T The Traces of Bessel Potentials on Regular Subsets of Carnot Groups
%J Matematičeskie trudy
%D 2007
%P 19-61
%V 10
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2007_10_2_a1/
%G ru
%F MT_2007_10_2_a1
S. K. Vodop'yanov; I. M. Pupyshev. The Traces of Bessel Potentials on Regular Subsets of Carnot Groups. Matematičeskie trudy, Tome 10 (2007) no. 2, pp. 19-61. http://geodesic.mathdoc.fr/item/MT_2007_10_2_a1/

[1] Besov O. V., “O nekotorom semeistve funktsionalnykh prostranstv. Teoremy vlozheniya i prodolzheniya”, DAN SSSR, 126 (1959), 1163–1165 | MR | Zbl

[2] Besov O. V., “Issledovanie odnogo semeistva funktsionalnykh prostranstv v svyazi s teoremami vlozheniya i prodolzheniya”, Tr. Mat. in-ta im. V. A. Steklova, 60, 1961, 42–81 | MR

[3] Besov O. V., “Povedenie differentsiruemykh funktsii na negladkoi poverkhnosti”, Tr. Mat. in-ta im. V. A. Steklova, 117, 1972, 3–10 | MR | Zbl

[4] Besov O. V., “O sledakh na negladkoi poverkhnosti klassov differentsiruemykh funktsii”, Tr. Mat. in-ta im. V. A. Steklova, 117, 1972, 11–21 | MR | Zbl

[5] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya., Nauka, Fizmatlit, M., 1996 | MR

[6] Vodopyanov S. K., “$L_p$-teoriya potentsiala i kvazikonformnye otobrazheniya na odnorodnykh gruppakh”, Sovremennye problemy geometrii i analiza, Nauka, Novosibirsk, 1989, 45–89 | MR

[7] Vodopyanov S. K., Kudryavtseva N. A., “Nelineinaya teoriya potentsiala dlya prostranstv Soboleva na gruppakh Karno”, Sib. mat. zhurn. (to appear)

[8] Vodopyanov S. K., Pupyshev I. M., “O granichnykh znacheniyakh differentsiruemykh funktsii, zadannykh v proizvolnoi oblasti gruppy Karno”, Dokl. RAN, 408:3 (2006), 295–300 | MR

[9] Vodopyanov S. K., Pupyshev I. M., “O granichnykh znacheniyakh differentsiruemykh funktsii, zadannykh v proizvolnoi oblasti gruppy Karno”, Mat. trudy, 9:2 (2006), 23–46 | MR

[10] Vodopyanov S. K., Pupyshev I. M., “Sledy prostranstv Soboleva na mnozhestvakh Alforsa grupp Karno”, Dokl. RAN, 411:2 (2006), 151–156 | MR

[11] Vodopyanov S. K., Pupyshev I. M., “Sledy funktsii iz prostranstv Soboleva na mnozhestvakh Alforsa grupp Karno”, Sib. mat. zhurn., 48:6 (2007), 1201–1221

[12] Vodopyanov S. K., Pupyshev I. M., “Teoremy tipa Uitni o prodolzhenii funktsii na gruppakh Karno”, Dokl. RAN, 406:5 (2006), 586–590 | MR

[13] Vodopyanov S. K., Pupyshev I. M., “Teoremy tipa Uitni o prodolzhenii funktsii na gruppakh Karno”, Sib. mat. zhurn., 47:4 (2006), 731–752 | MR

[14] Stein I. M., Singulyarnye integraly i differentsialnye svoistva funktsii, Mir, M., 1973 | MR

[15] Stein I. M., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[16] Capogna L. and Garofalo N., “Ahlfors regularity in Carnot–Carathéodory spaces”, J. Geom. Anal., 16:4 (2006), 455–497 | MR | Zbl

[17] Danielli D., Garofalo N., and Nhieu D.-M., Non-doubling Ahlfors Measures, Perimeter Measures, and the Characterization of the Trace Spaces of Sobolev Functions in Carnot–Carathéodory Spaces, Mem. Amer. Math. Soc., 182, no. 857, 2006 | MR

[18] Derridj M., “Sur un théorème de traces”, Ann. Inst. Fourier (Grenoble), 22:2 (1972), 73–83 | MR

[19] Folland G. B., “Subelliptic estimates and function spaces on nilpotent Lie groups”, Ark. Mat., 13 (1975), 161–207 | DOI | MR | Zbl

[20] Folland G. B. and Stein E. M., Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton, NJ, 1982 | MR | Zbl

[21] Jonsson A., “The trace of potentials on general sets”, Ark. Mat., 17 (1979), 1–18 | DOI | MR | Zbl

[22] Jonsson A. and Wallin H., “A Whitney extension theorem in $L_p$ and Besov spaces”, Ann. Inst. Fourier (Grenoble), 28 (1978), 139–192 | MR

[23] Jonsson A. and Wallin H., “Function Spaces on Subsets of $\mathbb R^n$”, Math. Rep. Ser. 2, 1984, no. 1, 1–121 | MR

[24] Stein E. M., “The characterization of functions arising as potentials. II”, Bull. Amer. Math. Soc., 68 (1962), 577–582 | DOI | MR | Zbl

[25] Whitney H., “Analytic extensions of differentiable functions defined in closed sets”, Trans. Amer. Math. Soc., 36 (1934), 63–89 | DOI | MR | Zbl