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Abstract

 - semigroup associated with singular Laplace- Bessel differential operator                  n n n  is introduced. Representations of the Riesz potentials        B ,   Re   via the  -semigroup are obtained. Mathematics Subject Classification: 26A33, 44A35

Keywords

• Riesz Potentials, Generalized Shift, Fourier- Bessel Transform, Laplace- Bessel Differential Operator.

ON  SEMIGROUP GENERATED BY FOURIER- BESSEL TRANSFORM AND RIESZ POTENTIAL ASSOCIATED WITH   SEMIGROUP

Abstract

- semigroup associated with singular Laplace- Bessel differential operator

 

is introduced. Representations of the Riesz potentials ,  via the -semigroup are obtained.

 

Mathematics Subject Classification: 26A33, 44A35

Keywords

• Genelleşmiş kayma operatörü, Riesz Potansiyeli, Fourier- Bessel dönüşümü, Laplace- Bessel diferansiyel operatörü.

References

1. Levitan, B. M., (1951), Bessel function expansions in series and Fourier Integrals, Uspekhi Mat. Nauk 6, No 2, 102-143 ( In Russian).
2. Stein, E. M., (1961), The characterization of functions arising as potentials, Bull. Amer. Math. Soc., 67, No 1, 101-104.
3. Muckenhoupt, B., Stein, E., (1965), Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Mat. Soc., 118, 17-92.
4. Kipriyanov, I. A., (1967), The Fourier Bessel Transformation and embedding theorems for weighted class, Trudy Mat. Inst. AN SSSR, 89, 130-213. ( In Russian)
5. Wheeden, R. L., (1969), On hypersingular Integrals and Lebesgue Spaces of differentiable functions, II. Trans. Amer. Math. Soc., 139, No 1, 37-53. Lizorkin, P. I., (1970), On description of   n r p R L space in the term of difference singular integrals, Math. Sbornik, 81, No 1, 79-91.
6. Stein, E. M., (1970), Singular integrals and differentiability Properties of Functions, Princeton Univ., Press, Princeton, N. J.
7. Kipriyanov, I. A., Klyuchantsev, M. I., (1970), On Singular Integrals generated by the generalized shift operator, II. Sibirsk. Mat. Zh. 11, 1060-1083 ( In Russian); English translate in: Siberian Math. J. 11
8. Klyuchantsev, M. I.,(1970), On Singular integrals generated by the generalized shift operator, I. Sibirsk. Mat. Zh. 11, 810-821 ( In Russian)

9. Fisher, M. J., (1971), Singular Integrals and fractional powers of operators, Trans. Amer. Math. Soc., 161, No 2, 307-326.
10. Samko, S. G. (1976), On Riesz potentials spaces, Izv. AN SSSR (Ser. Mat.), 40, No 5, 1143-1172.
11. Samko, S. G., (1977), The spaces   n r p R L  , and the hypersingular integrals, Stud. Math. (PRL), 61, No 3, 193-230.
12. Lyakhov, L. N., (1984), On classes of spherical functions and singular psedodifferential operators, Dokl. Akad Nauk 272, 781-784 ( In Russian)
13. Samko, S. G., Kilbas, A. A., Marichev, O.I., (1987), Integrals and derivatives of fractional order and some of their applications, Nauk. Tek., Minsk. Aliev, I. A., (1987), Riesz transform generated by a generalized shift operator, Izvestiya Akad Nauk Azaerbaijan Rep. ( Ser. Fiz. Techn. Math. Nauk) 1, 7-13 ( In Russian)
14. Gadjiev, A. D., Aliev, I. A., (1988), The Riesz and Bessel type potentails generated by a generalized shift operator and their inverse, Proc. IV All- Union Winter Conf., Theory of functions and Approximation, 47-53, Saratov.
15. Gadjiev, A. D., Aliev, I. A., (1988), On a class of potential type operator generated by a generalized shift operator, Reports Seminar of the I. N. Vekua Inst., Appl. Math., 3, No 2, Tbilisi.
16. Aliev, I. A., Gadjiev, A. D., (1994), Weighted estimates of multidimensional singular integrals generated by the generalized shift operator, Russian Acad. Sci. Sb. Math. 77, No 1, 37-55.
17. Rubin, B. S., (1996), Fractional Integrals and Potentials, Essex- U. K., Addison Wesley- Longman.
18. Guliyev, V. S. (1998), Sobolev’s theorem for Riesz B- Potentials, Dokl. Rus. Akad. Nauk, 358, No 4, 450-451.
19. Mourou, M. A., Trimeche, K., (1998), Calderon’s reproducing formula associated with the Bessel operator, J. Math. Anal. and Appl 219, 97
20. Bayrakci S., Aliev, I. A., (1998), On inversion of B- elliptic potentials by the method of Balakrishnan- Rubin, Fractioanl Calculus and Applies Analysis, Vol 1, No 4, 365-384.
21. Sezer, S., Aliev, I. A., (2010), A new characterization of the Riesz potential spaces with the aid of a composite wavelet transform, Journal of Mathematical Analysis and Applications, 372, 549-558. ****