International Journal of Mathematics Trends and Technology (IJMTT)
© 2021 by IJMTT Journal
Volume-67 Issue-3
Year of Publication : 2021


MLA Style: HASSEN BEN MOHAMED, MOHAMED MOKTAR CHAFFAR "GENERALIZED WEINSTEIN AND SOBOLEV SPACES" International Journal of Mathematics Trends and Technology 67.3 (2021):28-44. 

APA Style: HASSEN BEN MOHAMED, MOHAMED MOKTAR CHAFFAR(2021). GENERALIZED WEINSTEIN AND SOBOLEV SPACES International Journal of Mathematics Trends and Technology, 28-44.

In this paper we present a brief history and the basic ideas of the generalized Weinstein operator ΔWα,d,n which generalizes the Weinstein operator ΔWα,d,n. In n=0 we regain the Weinstein operator has several applications in pure and applied mathematics especially in fluid mechanics. We study the Sobolev spaces of exponential type Hsα,n(R+d+1) associated with the generalized Weinstein and investigate their properties, Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Using the theory of reproducing kernels ( which was written in 1942-1943), we introduce a class of symbols of exponential type and their associated pseudo differential operators related to the generalized Weinstein operator ΔWα,d,n and finally, we give some applications to these spaces.


[1] A. Aboulez, A. Achak, R. Daher and E. Loualid, Harmonic analysis associated with the generalized Weinstein operator. International.J. of Analysis and Applications. Vol.9,Nr 1, (2015), p. 19-28.
[2] H. Ben Mohamed, N. Bettaibi and S.H. Jah. Sobolev type spaces associated with the Weinstein operator , Int. Journal of Math. Analysis, Vol.5, Nr.28, (2011), p.1353- 1373.
[3] H. Ben Mohamed, B. Ghribi Weinstein-Sobolev spaces of exponential type and appli- cations . Acta Mathematica Sinica, English Series,Vol.29, Nr.3,(2013), p.591-608.
[4] H. Ben Mohamed, A. Gasmi and N. Bettaibi. Inversion of the Weinstein intertwining operator and its dual using Weinstein wavelets. An. St. Univ. Ovidius Constanta. Vol. 1, Nr1, (2016), p. 1-19.
[5] H. Ben Mohamed. On the Weinstein equations in spaces DPα,d type. International Journal of Open Problems in Complex Analysis. Vol.9, Nr 1, ( 2017), p. 39-59.
[6] K. El-Hussein. Fourier transform and Plancherel Theorem for Nilpotent Lie Group. International Journal of Mathematics Trends and Technology. Vol.4 Issue 11, (2013), p. 288-294.
[7] I. Aliev. Investigation on the Fourier-Bessel harmonic analusis, Doctoral Dissertation, Baku 1993 ( in Russian).
[8] I.A. Aliev and B. Rubin. Parabolic potentials and wavet transform with the generalized translation. Studia Math. 145 (2001) Nr1, p. 1-16.
[9] I.A. Aliev and B. Rubin. Spherical harmonics associated to the Laplace-Bessel operator and generalized spherical convolutions. Anal. Appl. (Singap) Nr 1 (2003), p. 81-109.
[10] S. Lee, Generalized Sobolev spaces of exponential type, Kangweon-Kyungki Math. J. Vol.8, Nr 1 (2000) p. 73-86.
[11] J. Lofstrom and J. Peetre. Approximation theorems connected with generalized translations. Math. Ann. Nr 181 (1969), p. 255-268.
[12] D. H. Pahk and B. H. Kang, Sobolev spaces in the generlized distribution spaces of Beurling type, Tsukuba J. Math. Vol. 15, (1991) p.325-334.
[13] R. S. Pathak, Generalized Sobolev spaces and pseudo-differential operators on spaces of ultradistributions, Structure of solutions of diffential equations, Edited y M.Morimoto and T. Kawai, World Scientific, Singafore (1996) p. 343-368.
[14] K. Stempak. La theorie de Littlewood-Paley pour la transformation de Fourier-Bessel. C.R. Acad.Sci. Paris Ser. I303 (1986), p. 15-18.
[15] K. Trimeche. Generalized Wavet and Hypergroups. Gordon and Breach, New York, 1997.
[16] A.Saoudi. A Variation of LP Uncertainty Principles in Weinstein Setting. Indian J.Pure Appl. Math., 51(4),(2020), p. 1697-1712.

Keywords : Sobolev Spaces, Generalized Weinstein operator, Generalized Weinstein transform, Weinstein, Kernel Reproducing Theory, pseudodifferential operator.