Opuscula Mathematica
Opuscula Math. 40, no. 2 (2020), 171-207

On the deformed Besov-Hankel spaces

Salem Ben Saïd
Mohamed Amine Boubatra
Mohamed Sifi
Abstract. In this paper we introduce function spaces denoted by \(BH_{\kappa,\beta}^{p,r}\) (\(0\lt\beta\lt 1\), \(1\leq p, r \leq +\infty\)) as subspaces of \(L^p\) that we call deformed Besov-Hankel spaces. We provide characterizations of these spaces in terms of Bochner-Riesz means in the case \(1\leq p\leq +\infty\) and in terms of partial Hankel integrals in the case \(1\lt p\lt +\infty\) associated to the deformed Hankel operator by a parameter \(\kappa\gt 0\). For \(p=r=+\infty\), we obtain an approximation result involving partial Hankel integrals.
Keywords: deformed Hankel kernel, Besov spaces, Bochner-Riesz means, partial Hankel integrals.
Mathematics Subject Classification: 44A15, 46E30.
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References ▼
  1. [1] B. Amri, A. Gasmi, M. Sifi, Linear and bilinear multiplier operators for the Dunkl transform, Mediterr. J. Math. 7 (2010), 503-521.
  2. [2] S. Ben Saïd, A Product formula and convolution structure for a k-Hankel transform on \(\mathbb{R}\), J. Math. Anal. Appl. 463 (2018) 2, 1132-1146.
  3. [3] J.J. Betancor, L. Rodríguez-Mesa, On the Besov-Hankel spaces, J. Math. Soc. Japan 50 (1998) 3, 781-788.
  4. [4] J.J. Betancor, L. Rodríguez-Mesa, Lipschitz-Hankel spaces and partial Hankel integrals, Integral Transforms Spec. Funct. 7 (1998) 1-2, 1-12.
  5. [5] J.J. Betancor, L. Rodríguez-Mesa, Lipschitz-Hankel spaces, partial Hankel integrals and Bochner-Riesz means, Arch. Math. 71 (1998) 2, 115-122.
  6. [6] M. Boureanu, V.D. Rădulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), 4471-4482.
  7. [7] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953.
  8. [8] A. Gasmi, M. Sifi, F. Soltani, Herz-type Hardy spaces for the Dunkl operator on the real line, Fract. Calc. Appl. Anal. 9 (2006) 3, 287-306.
  9. [9] D.V. Giang, F. Móricz, A new characterization of Besov spaces on the real line, J. Math. Anal. Appl. 189 (1995), 533-551.
  10. [10] J. Gosselin, K. Stempak, A weak-type estimate for Fourier-Bessel multipliers, Proc. Amer. Math. Soc. 106 (1989) 3, 655-662.
  11. [11] L. Kamoun, Besov-type spaces for the Dunkl operator on the real line, J. Comput. Appl. Math. 199 (2007), 56-67.
  12. [12] L. Kamoun, S. Negzaoui, Lipschitz spaces associated with reflection group \(\mathbb{Z}^d_2\), Commun. Math. Anal. 7 (2009) 1, 21-36.
  13. [13] M. Mihăilescu, V.D. Rădulescu, Neumann problems associated to nonhomogeneous differential operators in Orlicz-Sobolev spaces, Ann. Inst. Fourier 58 (2008), 2087-2111.
  14. [14] S.M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Die Grun. Der Math. Wiss. in Einze. Band 205, Springer, Berlin, Heidelberg, New York, 1975.
  15. [15] V.D. Rădulescu, D.D. Repovs, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CCR Press, Taylor and Francis Group, Boca Raton FL, 2015.
  16. [16] P.G. Rooney, On the \(\mathcal{Y}_{\nu}\) and \(\mathcal{H}_{\nu}\) transformations, Canad. J. Math. 32 (1980) 5, 1021-1044.
  17. [17] M. Rosler, Bessel-type signed hypergroups on \(\mathbb{R}\), Probability measures on groups and related structures, XI (Oberwolfach, 1994), 292-304, World Sci. Publ., River Edge, NJ, 1995.
  18. [18] G.N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1966.
About the authors ▼
Salem Ben Saïd
https://orcid.org/0000-0001-7577-5167
Department of Mathematical Science, College of Science, United Arab Emirates University, Al-Ain, Abu Dhabi, UAE
Mohamed Amine Boubatra
https://orcid.org/0000-0002-3595-7246
Université Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire d'Analyse Mathématique et Applications, LR11ES11, Campus Universitaire, 2092 El Manar I, Tunis, Tunisia
Mohamed Sifi (corresponding author)
https://orcid.org/0000-0003-0607-8303
Université Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire d'Analyse Mathématique et Applications, LR11ES11, Campus Universitaire, 2092 El Manar I, Tunis, Tunisia
About the article ▼
Communicated by Vicentiu D. Radulescu.
Received: 2019-11-11.
Revised: 2020-01-29.
Accepted: 2020-01-30.
Published online: 2020-03-09.
Cite this article as ▼
Salem Ben Saïd, Mohamed Amine Boubatra, Mohamed Sifi, On the deformed Besov-Hankel spaces, Opuscula Math. 40, no. 2 (2020), 171-207, https://doi.org/10.7494/OpMath.2020.40.2.171.
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