Compositions of Hadamard-type fractional integration operators and the semigroup property

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 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 ∞). The semigroup property and its generalizations are established for the generalized Hadamard-type fractional integration operators under consideration. Conditions are also given for the boundedness in X p c of these operators; they involve Kummer confluent hypergeometric functions as kernels.  2002 Elsevier Science (USA). All rights reserved.