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.
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