Abstract
For the Hardy space \(H^p(\mathbb{K }),\;\) \( 0<p\le 1\) and \(\mathbb K \) being the Laguerre hypergroup, we shall establish a Hardy’s type inequality associated with Laguerre Fourier transform for the strip \(\frac{Q}{2}(2-p)<\sigma < \frac{Q}{2}+\frac{p}{2}(N+1),\) where \( N=\left[Q\left(\frac{1}{p}-1\right)\right]\) is the greatest integer not exceeding \(Q\left(\frac{1}{p}-1\right)\) and \(Q \) is the homogenous dimension of \(\mathbb{K }.\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assal, M.: Pseudo-differential operators associated with Laguerre hypergroups. J. Comp. Appl. Math. 233, 617–620 (2009)
Assal, M., Rahmouni, A.: Hardy’s type inequality associated with Laguerre Fourier transform. In: Integral Transforms and Special Functions, iFirst, pp. 18. (2012)
Assal, M.: Hardy’s type inequality associated with the Hankel transform for overcritical exponent, Integr. Transf. Spec. F. 22(1), 45–50 (2011)
Bloom, W.R., Heyer, H.: Harmonic analysis of probability measures on hypergroups. In: Bauer, H., Kazdan, J.L., Zehnder, E. (eds.) De Gruyter Stud. Math., vol. 20. de Gruyter, Berlin (1994)
Coifman, R.R.: A real-variable characterization of \(H^p\). Studia Math. 51, 269–274 (1974)
Faraut, J., Harzallah, K.: Deux cours d’Analyse Harmonique, In: Ecole d’été d’Analyse Harmonique de Tunis, Birkhaüser (1984)
Fefferman, C., Stein, E.M., Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)
Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)
Garcia-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics, North Holland (1985)
Kanjin, Y.: On Hardy-type inequalities and Hankel transforms. Monatshefte für Math. 127, 311–319 (1999)
Kanjin, Y.: Hardy’s inequalities for Hermite and Laguerre expansions. Bull. Lond. Math. Soc. 29, 331–337 (1997)
Nessibi, M.M.: A local central limit theorem on the Laguerre hypergroup. J. Math. Anal. Appl. 354, 630–640 (2009)
Nessibi, M.M., Trimèche, K.: Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets. J. Math. Anal. Appl. 208, 337–363 (1997)
Radha, R.: Hardy type inequalities. Taiwan. J. Math. 4, 447–456 (2000)
Radha, R., Thangavelu, S.: Hardy’s inequalities for Hermite and Laguerre expansions. Proc. Am. Math. Soc 132(12), 3525–3536 (2004)
Satake, M.: Hardy’s inequalities for Laguerre expansions. J. Math. Soc. 52(1), 17–24 (2000)
Stein, E.M.: Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)
Taibleson, M.H., Weiss, G.: The Molecular Characterization of Certain Hardy Spaces. In: Astérisque, vol. 77. Société Math. de France, Paris (1980)
Thangavelu, S.: On regularity of twisted spherical means and special Hermite expansion. Proc. Ind. Acad. Sci. 103, 303–320 (1993)
Trimèche, K.: Generalized wavelets and hypergroups, Gordon and Beach Sci. Publ., Berkshire, (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miloud, A., Atef, R. An improved Hardy’s inequality associated with the Laguerre Fourier transform. Collect. Math. 64, 283–291 (2013). https://doi.org/10.1007/s13348-012-0069-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-012-0069-9
Keywords
- Hardy space
- Atomic decomposition
- Hardy’s type inequality
- Laguerre Fourier transform
- Laguerre hypergroup