HADAMARD-TYPE FRACTIONAL CALCULUS

Abstract

The paper is devoted to the study of fractional integra- tion and dierentiation on a finite interval ( a;b) of the real axis in the frame of Hadamard setting. The constructions under consider- ation generalize the modified integration R x a (t=x) f(t)dt=t and the modified dierentiation - + (- = xD; D = d=dx) with real , be- ing taken n times. Conditions are given for such a Hadamard-type fractional integration operator to be bounded in the space X p c(a;b) of Lebesgue measurable functions f on R+ = (0;1) such that Z b a jt c f(t)j p dt t < 1 (1 • p < 1); ess supatb(u c jf(t)j) < 1 (p = 1); for c 2 R = (i1;1), in particular in the space L p (0;1) (1 • p • 1). The existence almost everywhere is established for the corresponding Hadamard-type fractional derivative for a function g(x) such that x g(x) have - derivatives up to order n i 1 on (a;b) and - ni1 (x g(x)) is absolutely continuous on (a;b). Semigroup and reciprocal properties for the above operators are proved.