Mellin transform analysis and integration by parts for Hadamard-type fractional integrals

AbstractThis paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space Xpc of Lebesgue measurable functions f on R+=(0,∞) such that ∫0∞ucf(u)pduu<∞(1⩽p<∞),esssupu>0ucf(u)<∞(p=∞), for c∈R=(−∞,∞), in particular in the space Lp(0,∞) (1⩽p⩽∞). Formulas for the Mellin transforms of the four Hadamard-type fractional integral operators are established as well as relations of fractional integration by parts for them