On generalized distributions associated with singular partial differential operators

Abstract

In this paper, we discuss various properties of the Riemann-Liouville operator over the generalized distributions $\mathcal{D}^{'}_w([0,+\infty[\times\mathbb{R})$ and other spaces. Next, we examine some properties of the convolution of the generalized distributions on the space $\mathcal{D}^{'}_w([0,+\infty[\times\mathbb{R})$.

Keywords

• Riemann-Liouville operator
• generalized distributions
• partial differential operator

References

1. [1] C. Baccar, N. B. Hamadi and L. T. Rachdi, Inversion formulas for Riemann-Liouville transform and its dual associated with singular partial diferential operators, Int. J. Math. Math. Sci. 2006, 1–26, 2006.
2. [2] C. Baccar, N. B. Hamadi and L. T. Rachdi, Best approximation for Weierstrass transform connected with Riemann-Liouville operator, Commun. Math. Anal. 5 (1), 65-83, 2008.
3. [3] C. Baccar and L. T. Rachdi, Spaces of DLp type and convolution product associated with the Riemann-Liouville operator, Bull. Math. Anal. Appl. 1(3), 16–41, 2009.
4. [4] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur applicationa une transformation fonctionelle. In: Ninth Scandinavian Mathematical Congress. 345–366, 1938.
5. [5] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6, 351–407, 1966.
6. [6] A. Gasmi and A.EL. Garna, Properties of the linear multiplier operator for the Weinstein transform and applications, Electron. J. Differ. Equ. 2017, 1–18, 2017.
7. [7] K. Hleili, Calderóns reproducing formulas and extremal functions for the RiemannLiouville -multiplier operators, J. Pseudo-Differ. Oper. Appl. 9 (1), 125–141, 2018.
8. [8] K. Hleili, A Variation of uncertainty principles for the continuous wavelet transform connected with the RiemannLiouville operator, Afrika Matematika. 34 (4), pages 84, 2023.

9. [9] K. Hleili and S. Omri, An $L^p- L^q$ version of Miyachis theorem for the Riemann- Liouville operator, Indian Journal of Pure and Applied Mathematics. 46, 121–138, 2015.
10. [10] K. Hleili, S. Omri and L. T. Rachdi, Uncertainty principle for the Riemann-Liouville operator, Cubo. 13 (3), 91–115, 2011.
11. [11] L. Hörmander, Linear Partial Differential Operators, 116, Springer, Berlin, 1963.
12. [12] H. Komatsu, Ultradistributions, I: Structure theorems and a characterization. J. Fac. Sci. Tokyo (Ser. IA) 20, 25–105, 1973.
13. [13] R.S. Pathak, and A.B. Pandey, On Hankel transforms of ultradistributions, Appl. Anal. 20, 245–268, 1985.
14. [14] C. Roumieu, Ultra-distributions définies sur $\mathbb{R}^n$ et sur certaines classes de variétés différentiables, J. Anal. Math. 10, 153–192, 1962.
15. [15] H.M. Srivastava, S. Yadav and S.K. Upadhyay, The Weinstein transform associated with a family of generalized distributions, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117 (132), 1–32, 2023.
16. [16] S. K. Upadhyay and S. Yadav, Generalized Sobolev type Spaces Involving the Weinstein transform, Poincare Journal of Analysis and Applications. 10 (3), 1-18, 2023.
17. [17] K. Trimèche, Generalized Harmonic Analysis and Wavelet Packets, Ann. Probab. Gordon and Breach Publishing group, 2001.