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-C(P) of Lebesgue measurable functions f on R+ = (0, infinity) such that integral(0)(infinity)\u(c)f(u)\(p)du/u < infinity (1 less than or equal to p less than or equal to infinity), ess sup(u>0)[u(c)\f(u)\] < infinity (p = infinity), for c is an element of R = (-infinity, infinity), in particular in the space L-p (0, infinity) (1 less than or equal to p less than or equal to infinity). 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. (C) 2002 Elsevier Science (USA). All rights reserved.