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

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.