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Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 4, Pages 119–134 (Mi vmj749)  

On Hadamard and Hadamard-type directional fractional integro-differentiation in weighted Lebesgue spaces with mixed norm

M. U. Yakhshiboev

National University of Uzbekistan named after Mirzo Ulugbek, 4 University St., Student's campus, Tashkent 100174, Uzbekistan

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.

Key words: 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.

DOI: https://doi.org/10.46698/t4957-0399-9092-y

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UDC: 517.983
MSC: 26A33, 41A35, 46E30
Received: 18.05.2020

Citation: M. U. Yakhshiboev, “On Hadamard and Hadamard-type directional fractional integro-differentiation in weighted Lebesgue spaces with mixed norm”, Vladikavkaz. Mat. Zh., 22:4 (2020), 119–134

Citation in format AMSBIB
\Bibitem{Yak20}
\by M.~U.~Yakhshiboev
\paper On Hadamard and Hadamard-type directional fractional integro-differentiation in weighted Lebesgue spaces with mixed norm
\jour Vladikavkaz. Mat. Zh.
\yr 2020
\vol 22
\issue 4
\pages 119--134
\mathnet{http://mi.mathnet.ru/vmj749}
\crossref{https://doi.org/10.46698/t4957-0399-9092-y}


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