Geodesic
On Hadamard and Hadamard-type directional fractional integro-differentiation in weighted Lebesgue spaces with mixed norm
Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 4, pp. 119-134

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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. 
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     author = {M. U. Yakhshiboev},
     title = {On {Hadamard} and {Hadamard-type} directional fractional integro-differentiation in weighted {Lebesgue} spaces with mixed norm},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {119--134},
     publisher = {mathdoc},
     volume = {22},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2020_22_4_a9/}
}
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M. U. Yakhshiboev. On Hadamard and Hadamard-type directional fractional integro-differentiation in weighted Lebesgue spaces with mixed norm. Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 4, pp. 119-134. http://geodesic.mathdoc.fr/item/VMJ_2020_22_4_a9/