Geodesic
Nikol'skii--Besov and Lizorkin--Triebel spaces constructed on the base of the multidimensional Fourier--Bessel transform
Eurasian mathematical journal, Tome 2 (2011) no. 3, pp. 42-66.

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

In this paper we define the Nikol'skii–Besov and Lizorkin–Triebel spaces ($B$-Nikol'skii–Besov and $B$-Lizorkin–Triebel spaces) in the context of the Fourier–Bessel harmonic analysis. We establish some basic properties of the $B$-Nikol'skii–Besov and $B$-Lizorkin–Triebel spaces such as embedding theorems, the lifting property, and characterizing of the Bessel potentials in terms of the $B$-Lizorkin–Triebel spaces. We prove the inclusion and the density of the Schwartz space in the $B$-Nikol'skii–Besov and $B$-Lizorkin–Triebel spaces and prove an interpolation formula for these spaces by the real method. We also prove the Young inequality for the $B$-convolution operators in the $B$-Bessel potential spaces. Finally, we give some applications involving the Laplace–Bessel differential operator. 
@article{EMJ_2011_2_3_a2,
     author = {V. S. Guliyev and A. Serbetci and A. Akbulut and Y. Y. Mammadov},
     title = {Nikol'skii--Besov and {Lizorkin--Triebel} spaces constructed on the base of the multidimensional {Fourier--Bessel} transform},
     journal = {Eurasian mathematical journal},
     pages = {42--66},
     publisher = {mathdoc},
     volume = {2},
     number = {3},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a2/}
}
TY  - JOUR
AU  - V. S. Guliyev
AU  - A. Serbetci
AU  - A. Akbulut
AU  - Y. Y. Mammadov
TI  - Nikol'skii--Besov and Lizorkin--Triebel spaces constructed on the base of the multidimensional Fourier--Bessel transform
JO  - Eurasian mathematical journal
PY  - 2011
SP  - 42
EP  - 66
VL  - 2
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a2/
LA  - en
ID  - EMJ_2011_2_3_a2
ER  - 
%0 Journal Article
%A V. S. Guliyev
%A A. Serbetci
%A A. Akbulut
%A Y. Y. Mammadov
%T Nikol'skii--Besov and Lizorkin--Triebel spaces constructed on the base of the multidimensional Fourier--Bessel transform
%J Eurasian mathematical journal
%D 2011
%P 42-66
%V 2
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a2/
%G en
%F EMJ_2011_2_3_a2
V. S. Guliyev; A. Serbetci; A. Akbulut; Y. Y. Mammadov. Nikol'skii--Besov and Lizorkin--Triebel spaces constructed on the base of the multidimensional Fourier--Bessel transform. Eurasian mathematical journal, Tome 2 (2011) no. 3, pp. 42-66. http://geodesic.mathdoc.fr/item/EMJ_2011_2_3_a2/

[1] I. A. Aliev, S. Uyhan-Bayrakci, “On inversion of Bessel potentials associated with the Laplace–Bessel differential operator”, Acta Math. Hungar., 95:1–2 (2002), 125–145 | DOI | MR | Zbl

[2] G. Altenburg, “Eine Realisierung der Theorie der abstrakten Besov–Raume $B^s_q(A)$, $s>0$ $(1\leq q\leq\infty)$ und der Lebesgue–Raume $H_{p,\gamma}^s$ auf der Grundlage Besselscher Differential-operatoren”, Z. Anal. Anwendungen, 3:1 (1984), 43–63 | MR | Zbl

[3] M. Assal, H. Ben Abdallah, “Generalized weighted Besov spaces on the Bessel hypergroup”, J. Funct. Spaces Appl., 4:1 (2006), 91–111 | DOI | MR | Zbl

[4] J. Bergh, J. Löfstrom, Interpolation spaces, an introduction, Springer-Verlag, Berlin–Heidelberg, 1976 | MR | Zbl

[5] J. J. Betancor, L. Rodríguez-Mesa, “On Besov spaces in the Hankel setting”, Acta Math. Hungar., 111:3 (2006), 237–262 | DOI | MR | Zbl

[6] D. I. Cruz-Baez, J. Rodriguez, “Spaces of distributions of Besov and Lizorkin–Triebel type for the Fourier-Bessel transform”, Journal of Mathematical Analysis and Applications, 259 (2001), 51–63 | DOI | MR | Zbl

[7] A. D. Gadjiev, I. A. Aliev, “Riesz and Bessel potentials generated by the generalized shift operator and its inversions”, Theory of Functions and Approximation, Saratov Univ., Saratov, 1990, 47–53 | MR

[8] V. S. Guliev, “Sobolev theorems for $B$-Riesz potentials”, Dokl. RAS, 358:4 (1998), 450–451 (in Russian) | MR | Zbl

[9] B. M. Levitan, “Bessel function expansions in series and Fourier integrals”, Uspekhi Mat. Nauk, 6:2 (1951), 102–143 (in Russian) | MR | Zbl

[10] J. Löfstrom, J. Peetre, “Approximation theorems connected with generalized translations”, Math. Ann., 181 (1969), 255–268 | DOI | MR | Zbl

[11] L. N. Lyakhov, “Spaces of Riesz $B$-potentials”, Dokl. Akad. Nauk, 334:3 (1994), 278–280 (in Russian) | MR | Zbl

[12] L. N. Lyakhov, “Multipliers of the mixed Fourier–Bessel transformation”, Proc. V. A. Steklov Inst. Math., 214, 1997, 234–249 | MR | Zbl

[13] I. A. Kipriyanov, “Fourier–Bessel transformations and imbedding theorems”, Trudy Math. Inst. Steklov, 89, 1967, 130–213 | MR | Zbl

[14] R. S. Pathak, P. K. Pandey, “Sobolev Type Spaces Associated with Bessel Operators”, Journal of Mathematical Analysis and Applications, 215 (1997), 95–111 | DOI | MR | Zbl

[15] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978 | MR | Zbl

[16] M. H. Taibleson, “On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I”, J. Math. Mech., 13 (1964), 407–479 | MR

[17] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970 | MR

[18] A. Zygmund, Trigonometric Series, 2nd ed., Cambridge Univ. Press, Cambridge, 1959 | MR | Zbl

[19] S. Uyhan, “On Poisson semigroup generated by the generalized $B$-translation”, Dynamical Systems and Applications, Proceedings International Conference, 2004, 679–686 http://faculty.uaeu.ac.ae/hakca/papers/uyhan.pdf