Fractional differentiation of the Grunwald-Letnikov-Hadamard type and the difference of the fractional order with a multiplicative step

The properties of \u27\u27convolution-type\u27\u27 operators that are invariant with respect to dilation and to their approximation using a unity in weighted mixed Lebesgue spaces are studied in this paper. Integral representations are obtained for the Marchaud-Hadamard and Marchaud-Hadamard type truncated fractional derivatives (by direction and mixed ones). This paper introduces the concept of a mixed difference of a vector fractional order with a multiplicative step and its properties. Some of these properties are proven using the Mellin transform. In this paper, we give the proof of theorems on coincidence of the definition domains of two different forms of fractional differentiation operators of the Marchaud-Hadamard and Grunwald-Letnikov-Hadamard type (by direction and mixed ones) in weighted mixed Lebesgue spaces. In addition, the necessary and sufficient conditions for the existence of a fractional derivative of the Marchaud-Hadamard type by direction ω are obtained