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

• Published in: Journal of mathematical analysis and applications Vol. 270; no. 1; pp. 1 - 15
• Main Authors: Butzer, Paul L., Kilbas, Anatoly A., Trujillo, Juan J.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.06.2002; Elsevier
• Subjects: Exact sciences and technology; Fractional integration by parts; Hadamard-type fractional integration; Integral transforms, operational calculus; Mathematical analysis; Mathematics; Mellin transforms; Operator theory; Sciences and techniques of general use; Spaces of p-summable functions; Hadamard-type fractional integration; Fractional integration by parts; Mellin transforms; Spaces of p-summable functions; Integral operator; Measurable function; Function space; P summable function; Fractional integral; Integration by parts; Hadamard transformation; Lebesgue integral; Operator theory; Mellin transformation
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/S0022-247X(02)00066-5

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 X p c of Lebesgue measurable functions f on R +=(0,∞) such that ∫ 0 ∞ u cf(u) p du u <∞(1⩽p<∞), ess sup u>0 u c f(u) <∞(p=∞), for c∈ R =(−∞,∞) , in particular in the space L p (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.