Generalized Besov type spaces on the Laguerre hypergroup. (English) Zbl 1098.42016

Summary: 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.


42B35 Function spaces arising in harmonic analysis
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI Numdam EuDML


[1] Assal, M.; Nessibi, M. M., Sobolev Type Spaces on the Dual of the Laguerre Hypergroup, Potential Analysis, 20, 85-103 (2004) · Zbl 1060.42010 · doi:10.1023/A:1025542009772
[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, 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, 60, 42-81 (1961) · Zbl 0158.13901
[4] Betancor, J. J.; Rodríguez-Mesa, L., On the Besov-Hankel Spaces, Math. Soc. Japan, 50 n ∘ 3, 781-788 (1998) · Zbl 0904.46022 · doi:10.2969/jmsj/05030781
[5] Bloom, W. R.; Heyer, H., Harmonic Analysis of Probability Measures on Hypergroups (1994) · Zbl 0828.43005
[6] Bourdaud, G., Réalisation des Espaces de Besov Homogènes, Arkiv for Mat., 26, 41-54 (1988) · Zbl 0661.46026 · doi:10.1007/BF02386107
[7] Bourdaud, G., Analyse Fonctionnelle dans l’Espace Euclidien (1995) · Zbl 0627.46048
[8] Chemin, J.-Y, About Navier-Stokes System (1996)
[9] Coulhon, T.; Russ, E.; Tardivel-Nachef, V., Sobolev Algebras on Lie Groups and Riemannian Manifolds, Amer. J. of Math., 123, 283-342 (2001) · Zbl 0990.43003 · doi:10.1353/ajm.2001.0009
[10] Delsarte, J., Sur une Extension de la Formule de Taylor, J. Math. et Appl., 17, Fasc. III, 213-231 (1938) · JFM 64.0405.03
[11] Erdélyi, A.; Magnus, W.; Aberthettinger, F.; Tricomi, G., Higher Transcendental Functions. Vol II (1993) · Zbl 0052.29502
[12] Faraut, J.; Harzallah, K., Deux cours d’Analyse Harmonique, Ecole d’été d’Analyse Harmonique de Tunis (1984) · Zbl 0622.43001
[13] Fitouhi, A.; Hamza, M. M., Expansion In Series of Laguerre Functions for Solution of Perturbed Laguerre Equations · 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, 79 (1991)
[15] Jewett, R. I., Spaces with an Abstract Convolution of Measures, Adv. in Math., 18, 1-101 (1975) · Zbl 0325.42017 · doi:10.1016/0001-8708(75)90002-X
[16] Koornwinder, T. H., Positivity Proofs for Linearisation and Connection Coefficients of Orthogonal Polynomials Satisfying an Addition Formula, J. London. Math. Soc., 18, 101-1114 (1978) · Zbl 0386.33009 · doi:10.1112/jlms/s2-18.1.101
[17] Lebedev, N. M., Special Function and Their Applications (1972) · 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, 133-145 (1998) · 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, 208, 337-363 (1997) · Zbl 0870.43004 · doi:10.1006/jmaa.1997.5299
[20] Peetre, J., New Thoughts on Besov Spaces, Duke Univ. Math. Serie Durham, Univ. (1976) · Zbl 0356.46038
[21] Stein, E. M., Harmonic Analysis: Real Variable Methods, Orthogonality and Ossillatory Integrals (1993) · Zbl 0821.42001
[22] Stempak, K., An Algebra Associated With the Generalized Sublaplacian, Studia Math., 88, 245-256 (1988) · Zbl 0672.46025
[23] Stempak, K., Mean of Summability Methods for Laguerre Series, Transactions of the American mathematical society, 322, Number 2 (1990) · Zbl 0713.42024
[24] Triebel, H., Theory of Function Spaces (1983) · Zbl 0546.46028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.