Generalized Besov type spaces on the Laguerre hypergroup
Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 1, p. 117-145

In this paper we study generalized Besov type spaces on the Laguerre hypergroup and we give some characterizations using different equivalent norms which allows to reach results of completeness, continuous embeddings and density of some subspaces. A generalized Calderón-Zygmund formula adapted to the harmonic analysis on the Laguerre Hypergroup is obtained inducing two more equivalent norms.

@article{AMBP_2005__12_1_117_0,
     author = {Assal, Miloud and Ben Abdallah, Hacen},
     title = {Generalized Besov type spaces on the Laguerre hypergroup},
     journal = {Annales math\'ematiques Blaise Pascal},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {12},
     number = {1},
     year = {2005},
     pages = {117-145},
     doi = {10.5802/ambp.198},
     mrnumber = {2126444},
     zbl = {1098.42016},
     language = {en},
     url = {http://www.numdam.org/item/AMBP_2005__12_1_117_0}
}
Assal, Miloud; Ben Abdallah, Hacen. Generalized Besov type spaces on the Laguerre hypergroup. Annales mathématiques Blaise Pascal, Volume 12 (2005) no. 1, pp. 117-145. doi : 10.5802/ambp.198. http://www.numdam.org/item/AMBP_2005__12_1_117_0/

[1] Assal, M.; Nessibi, M. M. Sobolev Type Spaces on the Dual of the Laguerre Hypergroup, Potential Analysis, Tome 20 (2004), pp. 85-103 | Article | MR 2032613 | Zbl 1060.42010

[2] Bahouri, H.; Gérard, P.; Xu, C. J. Espaces de Besov et Estimations de Strichartz Généralisées sur le Groupe de Heisenberg, Journal d’Analyse Mathématique, Tome 82 (2000) | Zbl 0965.22010

[3] Besov, O. V. On a Family of Function Spaces in Connection with Embeddings and Extensions, Trudy Mat. Inst. Steklov, Tome 60 (1961), pp. 42-81 | MR 133675 | Zbl 0158.13901

[4] Betancor, J. J.; Rodríguez-Mesa, L. On the Besov-Hankel Spaces, Math. Soc. Japan, Tome 50 n ∘ 3 (1998), pp. 781-788 | Article | MR 1626374 | Zbl 0904.46022

[5] Bloom, W. R.; Heyer, H. Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter studies in Mathematics 20, de Gruyter, Berlin / New York (1994) | MR 1312826 | Zbl 0828.43005

[6] Bourdaud, G. Réalisation des Espaces de Besov Homogènes, Arkiv for Mat., Tome 26 (1988), pp. 41-54 | Article | MR 948279 | Zbl 0661.46026

[7] Bourdaud, G. Analyse Fonctionnelle dans l’Espace Euclidien, Publications Mathématiques de Paris 7 (1995) | Zbl 0627.46048

[8] Chemin, J.-Y About Navier-Stokes System, Publication du Laboratoire d’Analyse Numerique, Université Pierre et Marie Curie (1996)

[9] Coulhon, T.; Russ, E.; Tardivel-Nachef, V. Sobolev Algebras on Lie Groups and Riemannian Manifolds, Amer. J. of Math., Tome 123 (2001), pp. 283-342 | Article | MR 1828225 | Zbl 0990.43003

[10] Delsarte, J. Sur une Extension de la Formule de Taylor, J. Math. et Appl., Tome 17, Fasc. III (1938), pp. 213-231

[11] Erdélyi, A.; Magnus, W.; Aberthettinger, F.; Tricomi, G. Higher Transcendental Functions. Vol II, M.G Graw-Hill, New York (1993) | Zbl 0052.29502

[12] Faraut, J.; Harzallah, K. Deux cours d’Analyse Harmonique, Ecole d’été d’Analyse Harmonique de Tunis, Birkhäuser (1984) | Zbl 0622.43001

[13] Fitouhi, A.; Hamza, M. M. Expansion In Series of Laguerre Functions for Solution of Perturbed Laguerre Equations (to appear in Contemporary Mathematics of A.M.S.) | Zbl 0919.33004

[14] Frazier, M.; Jawerth, B.; Weiss, G. Sur une Extension de la Formule de Taylor, C.B.M.S, Amer. Math. Soc., Providence, Rhod Island, Tome 79 (1991) | MR 1107300

[15] Jewett, R. I. Spaces with an Abstract Convolution of Measures, Adv. in Math., Tome 18 (1975), pp. 1-101 | Article | MR 394034 | Zbl 0325.42017

[16] Koornwinder, T. H. Positivity Proofs for Linearisation and Connection Coefficients of Orthogonal Polynomials Satisfying an Addition Formula, J. London. Math. Soc., Tome 18 (1978), pp. 101-1114 | Article | MR 493144 | Zbl 0386.33009

[17] Lebedev, N. M. Special Function and Their Applications, Dover, New York (1972) | MR 350075 | Zbl 0271.33001

[18] Nessibi, M. M.; Sifi, M. Laguerre Hypergroup and Limit Theorem, B.P. Komrakov, I.S. Krasil’shchik, G.L. Litvinov and A.B. Sossinsky (eds.), Lie Groups and Lie Algebra-Their Representations, Generalizations and Applications, Kluwer Acad. Publ. Dordrecht (1998), pp. 133-145 | Zbl 0905.60016

[19] Nessibi, M. M.; Trimèche, K. Inversion of the Radon Transform on the Laguerre Hypergroup by Using Generalized Wavelets, Journal of mathematical analysis and application, Tome 208 (1997), pp. 337-363 | Article | MR 1441440 | Zbl 0870.43004

[20] Peetre, J. New Thoughts on Besov Spaces, Duke Univ. Math. Serie Durham, Univ. (1976) | MR 461123 | Zbl 0356.46038

[21] Stein, E. M. Harmonic Analysis: Real Variable Methods, Orthogonality and Ossillatory Integrals, Princeton University Press (1993) | MR 1232192 | Zbl 0821.42001

[22] Stempak, K. An Algebra Associated With the Generalized Sublaplacian, Studia Math., Tome 88 (1988), pp. 245-256 | MR 932012 | Zbl 0672.46025

[23] Stempak, K. Mean of Summability Methods for Laguerre Series, Transactions of the American mathematical society, Tome 322, Number 2 (December 1990) | MR 974528 | Zbl 0713.42024

[24] Triebel, H. Theory of Function Spaces, Birkhäuser (1983) | MR 781540