Abstract: The paper presents definitions and various auxiliary properties of Hadamard and Hadamard-type directional fractional integrals, Marchaud-Hadamard and Marchaud-Hadamard-type directional fractional derivatives. A relation is established between Hadamard and Hadamard-type directional fractional integrals and Marchaud-Hadamard and Marchaud-Hadamard-type directional fractional derivatives with the directional Riemann-Liouville operator. A modification of Hadamard and Hadamard-type directional fractional integrals with the kernel improved at infinity is introduced. The paper deals with a stretch invariant "convolution type" operators in weighted Lebesgue spaces with mixed norm. The boundedness and semigroup properties of Hadamard and Hadamard-type directional fractional integration in weighted Lebesgue spaces with mixed norm are proved. The compositions of Hadamard and Hadamard-type fractional integral and Marchaud-Hadamard and Marchaud-Hadamard-type directional fractional derivative are also considered and integral representation of Marchaud-Hadamard and Marchaud-Hadamard-type truncated directional fractional derivatives is obtained. Inversion theorems are proved for Hadamard and Hadamard-type directional fractional integrals on weighted Lebesgue spaces with mixed norm. A relationship between ordinary and truncated Marchaud-Hadamard and Marchaud-Hadamard-type directional fractional derivatives is also revealed.
Keywords: Hadamard fractional integral, Hadamard fractional derivative, Lebesgue space with mixed norm, dilation operator, fractional derivative by direction of the Marshau--Hadamard, fractional derivative by~direction of the Marshau-Hadamard type
For citation: Yakhshiboev, M. U. On Hadamard and Hadamard-Type Directional Fractional Integro-Differentiation in Weighted Lebesgue Spaces with Mixed Norm,
Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp.119-134 (in Russian). DOI 10.46698/t4957-0399-9092-y
1. Hadamard, J. Essai sur l'Etude des Functions Donnees par Leur
Developpment de Taylor, Journal de Mathematiques Pures et Appliquees, 1892, vol. 8, no. 4, pp. 101-186.
2. Benedek, A. and Panzone, R. The Space \(L^{p}\), with Mixed Norm,
Duke Mathematical Journal, 1961, vol. 28, no. 3, pp. 301-324. DOI: 10.1215/s0012-7094-61-02828-9.
3. Antonic, N. and Ivec, I. On the Hormander-Mihlin Theorem for Mixed-norm
Lebesgue Spaces, Journal of Mathematical Analysis and Applications, 2016, vol. 433, no. 1, pp. 176-199.
DOI: 10.1112/S0024610704005502.
4. Stefanov, A. and Torres, R. H. Calderon-Zygmund Operators on Mixed Lebesgue Spaces and Applications to Null Forms, Journal of the London Mathematical Society, 2004, vol. 70, no. 2, pp. 447-462. DOI.org/10.1112/S0024610704005502.
5. Besov, O. V., Il'in, V. P. and Nikol'skii, S. M.
Integral'nye predstavleniya funkcij i teoremy vlozheniya
[Integral Representations of Functions, and Embedding Theorems],
Moscow, Nauka, 1975, 480 p. (in Russian).
6. Marchaud A. P. Sur les Derivees et sur les Differences des Functions
de Variables Reelles, Journal de Mathematiques Pures et Appliquees,
1927, vol. 6, pp. 337-426.
7. Kipriyanov, I. A.
The Operator of Fractional Differentiation and the Powers of Elliptic
Operators, Dokl. Akad. Nauk SSSR, 1960, vol. 131, no. 2, pp. 238-241 (in Russian).
8. Kipriyanov, I. A. On Spaces of Fractional-Differentiable Functions,
Izv. Akad. Nauk SSSR. Ser. Mat., 1960, vol. 24, no. 6, pp. 865-882 (in Russian).
9. Kukushkin, M. V. On some Qualitative Properties of the Operator of Fractional
Differentiation in Kipriyanov Sense, Vestnik SamU. Estestvenno-Nauchnaya Ser.,
2017, no. 2, pp. 32-43 (in Russian).
10. Butzer, P. L., Kilbas, A. A. and Trujillo, J. J. Fractional Calculus
in the Mellin Setting and Hadamard-Type Fractional Integrals,
Journal of Mathematical Analysis and Applications, 2002, vol. 269, no. 1,
pp. 1-27. DOI: 10.1016/S0022-247X(02)00049-5.
11. Kilbas, A. A. Hadamard-type Fractional Calculus, Journal of the Korean
Mathematical Society, 2001, vol. 38, no. 6, pp. 1191-1204.
12. Kilbas, A. A. and Titioura, A. A. A Marchaud-Hadamard-Type Fractional Derivatives and Inversion of Hadamard-Type Fractional Integrals, Report of the Academician of Sciences of Belarus, 2006, vol. 50, no. 4, pp. 10-15 (in Russian).
13. Ma, L. and Li, C. On Hadamard Fractional Calculus, World Scientific, Fractals, 2017, vol. 25, no. 3, pp. 1-13. DOI: 10.1142/S0218348X17500335.
14. Samko, S. G. and Yakhshiboyev, M. U. A Chen-Type Modification of Hadamard
Fractional Integro-Differentiation, Operator Theory: Advances and Applications, 2014, vol. 242, pp. 325-339.
15. Wu, Y., Yao, K. and Zhang, X. The Hadamard Fractional Calculus of a Fractal Function,
World Scientific, Fractals, 2018, vol. 26, no. 3, pp. 25-36. DOI: 10.1142/s0218348x18500251.
16. Yakhshiboev, M. U. Hadamard-Type Fractional Integrals and Marchaud-Hadamard-Type
Fractional Derivatives in the Spaces With Power Weight, Uzbek Mathematical Journal,
2019, no. 3, pp. 155-174. DOI: 10.29229/uzmj.2019-3-17.
17. Samko, S. G., Kilbas, A. A. and Marichev, O. I.
Integraly i proizvodnye drobnogo poryadka i nekotorye ih prilozheniya [Fractional Integrals
and Derivatives and some of their Applications], Minsk, Nauka and Tekhnika, 1987, 688 p. (in Russian).
18. Berdyshev, A. S., Turmetov, B. Kh. and Kadirkulov, B. J. Some Properties and Applications of the Integrodifferential Operators of Hadamard-Marchaud Type in the Class of Harmonic Functions, Siberian Mathematical Journal, 2012, vol. 53, no. 4, pp. 600-612. DOI: 10.1134/S0037446612040039.
19. Kilbas, A. A. and Titioura, A. A. Nonlinear Differential Equation with Marchaud-Hadamard-Type
Fractional Derivative in the Weighted Space of Summable Functions, Mathematical Modelling
and Analysis, 2007, vol. 12, no. 3, pp. 343-356. DOI: 10.3846/1392-6292.2007.12.343-356.
20. Stein, E. M. Singulyarnye integraly i differencial'nye svojstva funkcij
[Singular Integrals and Differentialiability Properties of Functions], Moscow,
Mir, 1973, 342 p. (in Russian).
