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
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