Fractional calculus in the Mellin setting and Hadamard-type fractional integrals

Abstract

The purpose of this paper and some to follow is to present a new approach to fractional integration and differentiation on the half-axis R +=(0,∞) in terms of Mellin analysis. The natural operator of fractional integration in this setting is not the classical Liouville fractional integral I α 0+ f but J α 0+,cf (x):= 1 Γ(α) ∫ 0 x u x c log x u α−1 f(u) du u (x>0) for α>0, c∈ R . The Mellin transform of this operator is simply (c−s) −α M[f](s) , for s= c+ it, c,t∈ R . The Mellin transform of the associated fractional differentiation operator D α 0+,cf is similar: (c−s) α M[f](s) . The operator D α 0+,cf may even be represented as a series in terms of x k f ( k) ( x), k∈ N 0 , the coefficients being certain generalized Stirling functions S c ( α, k) of second kind. It turns out that the new fractional integral J α 0+,cf and three further related ones are not the classical fractional integrals of Hadamard (J. Mat. Pure Appl. Ser. 4, 8 (1892) 101–186) but far reaching generalizations and modifications of these. These four new integral operators are first studied in detail in this paper. More specifically, conditions will be given for these four operators to be bounded in the space X c p of Lebesgue measurable functions f on (0,∞), for c∈(−∞,∞), such that ∫ ∞ 0|u cf(u)| p du/u<∞ for 1⩽ p<∞ and ess sup u>0[u c|f(u)|]<∞ for p=∞, in particular in the space L p (0,∞) for 1⩽ p⩽∞. Connections of these operators with the Liouville fractional integration operators are discussed. The Mellin convolution product in the above spaces plays an important role.