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

• Published in: Journal of mathematical analysis and applications Vol. 269; no. 2; pp. 387 - 400
• Main Authors: Butzer, Paul L., Kilbas, Anatoly A., Trujillo, Juan J.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 15.05.2002; Elsevier
• Subjects: Confluent hypergeometric functions; Exact sciences and technology; Hadamard-type fractional integration; Integral transforms, operational calculus; Mathematical analysis; Mathematics; Operator theory; Sciences and techniques of general use; Spaces of p-summable functions; Special functions; Hadamard-type fractional integration; Spaces of p-summable functions; Confluent hypergeometric functions; Kernel function; Integral operator; P summable function; Fractional integral transform; Hypergeometric function; Bounded operator; Hadamard transformation; Semigroup; Special function; Operator theory; Kummer function
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/S0022-247X(02)00049-5

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 Xcp of functions f on R+=(0,∞) such that ∫0∞ucf(u)pduu<∞(1⩽p<∞),esssupu>0uc|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 Xcp of these operators; they involve Kummer confluent hypergeometric functions as kernels.