On boundedness of fractional Hadamard integration and Hadamard-type integration in Lebesgue spaces with mixed norm
Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 178-189.

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In this paper, we consider the boundedness of integrals of fractional Hadamard integration and Hadamard-type integration (mixed and directional) in Lebesgue spaces with mixed norm. We prove Sobolev-type theorems of boundedness of one-dimensional and multidimensional Hadamard-type fractional integration in weighted Lebesgue spaces with mixed norm.
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M. U. Yakhshiboyev. On boundedness of fractional Hadamard integration and Hadamard-type integration in Lebesgue spaces with mixed norm. Contemporary Mathematics. Fundamental Directions, Science — Technology — Education — Mathematics — Medicine, Tome 68 (2022) no. 1, pp. 178-189. http://geodesic.mathdoc.fr/item/CMFD_2022_68_1_a13/

[1] Berdyshev A. S., Turmetov B. Kh., Kadirkulov B. I., “Nekotorye svoistva i primeneniya integrodifferentsialnykh operatorov tipa Adamara—Marsho klasse garmonicheskikh funktsii”, Sib. mat. zh., 53:4 (2012), 752–764 | MR | Zbl

[2] Besov O. V., Ilin V. P., Nikolskii S. M., Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1975 | MR

[3] Kilbas A. A., Tityura A. A., “Drobnye integraly i proizvodnye tipa Adamara”, Tr. in-ta mat., 11 (2002), 79–87

[4] Lizorkin P. I., “Multiplikatory integralov Fure i otsenki svertok v prostranstvakh so smeshannoi normoi. Prilozheniya”, Izv. AN SSSR. Ser. mat., 34:1 (1970), 218–247 | Zbl

[5] Samko S. G., Gipersingulyarnye integraly i ikh prilozheniya, Rostovskii un-t, Rostov-na-Donu, 1984 | MR

[6] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987

[7] Antonic N., Ivec I., “On the Hormander—Mihlin theorem for mixed-norm Lebesgue spaces”, J. Math. Anal. Appl., 433 (2016), 176–199 | DOI | MR | Zbl

[8] Benedek A., Calderon A. P., Panzone R., “Convolution operators on Banach space valued functions”, Proc. Natl. Acad. Sci. USA, 48 (1962), 356–365 | DOI | MR | Zbl

[9] Benedek A., Panzone R., “The space $L^{p}$ with mixed norm”, Duke Math. J., 28 (1961), 301–324 | DOI | MR | Zbl

[10] Butzer P. L., Kilbas A. A., Trujillo J. J., “Fractional calculus in the Mellin setting and Hadamard-type fractional integrals”, J. Math. Anal. Appl., 269:1 (2002), 1–27 | DOI | MR | Zbl

[11] Butzer P. L., Kilbas A. A., Trujillo J. J., “Compositions of Hadamard-type fractional integration operators and the semigroup property”, J. Math. Anal. Appl., 269:2 (2002), 387–400 | DOI | MR | Zbl

[12] Butzer P. L., Kilbas A. A., Trujillo J. J., “Mellin transform analysis and integration by parts for Hadamard-type fractional integrals”, J. Math. Anal. Appl., 270:1 (2002), 1–15 | DOI | MR | Zbl

[13] Fernandez D. L., “Vector-valued singular integral operators on $L^{p}$-spaces with mixed norms and applications”, Pacific J. Math., 129:2 (1987), 257–275 | DOI | MR | Zbl

[14] Hadamard J., “Essai sur l`etude des functions données par leur développment de Taylor”, J. Math. Pures Appl., 8:4 (1892), 101–186

[15] Kenig C. E., Ponce G., Vega L., “Well-posedness and scattering results for the generalized Korteweg—de Vries equation via the contraction principle”, Commun. Pure Appl. Math., 46 (1993), 527–620 | DOI | MR | Zbl

[16] Kilbas A., “Hadamard-type fractional calculus”, J. Korean Math. Soc., 38:6 (2001), 1191–1204 | MR | Zbl

[17] Kilbas A., “Hadamard-type integral equations and fractional calculus operators”, Singular integral operators, factorization and applications, Oper. Theory Adv. Appl., 142, 2003, 175–188 | MR | Zbl

[18] Kim D., “Elliptic and parabolic equations with measurable coefficients in Lp-spaces with mixed norms”, Methods Appl. Anal., 15 (2008), 437–468 | DOI | MR | Zbl

[19] Samko S. G., Yakhshiboyev M. U., “A Chen-type modification of Hadamard fractional integro-dfferentiation”, Oper. Theory Adv. Appl., 242, 2014, 325–339 | MR | Zbl

[20] Stefanov A., Torres R. H., “Calderon—Zygmund operators on mixed Lebesgue spaces and applications to null forms”, J. London Math. Soc., 2:70 (2004), 447–462 | DOI | MR | Zbl

[21] Yakhshiboev M. U., “Hadamard-type fractional integrals and Marchaud—Hadamard-type fractional derivatives in the spaces with power weight”, Uzbek Math. J., 2019, no. 3, 155–174 | DOI | MR | Zbl