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

This paper is devoted to the study of four integral operators that are basic generalizations and modifications of fractional integrals of Hadamard, in the space X p c of Lebesgue measurable functions f on R+ = (0,∞) such that ∞ 0 ucf (u) p du u <∞ (1 p <∞), ess sup u>0 uc f (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.  2002 Elsevier Science (USA). All rights reserved.