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Hadamard-type fractional calculus. (English) Zbl 1018.26003

Summary: The paper is devoted to the study of fractional integration and differentiation on a finite interval \([a,b]\) of the real axis in the frame of Hadamard setting. The constructions under consideration generalize the modified integration \(\int^x_a(t/x)^\mu f(t) dt/t\) and the modified differentiation \(\delta+ \mu\) (\(\delta= xD\), \(D= d/dx\)) with real \(\mu\), being 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 \(\mathbb{R}_+= (0,\infty)\) such that \[ \int^b_a|t^c f(t)|^p {dt\over t}< \infty\qquad (1\leq p< \infty), \]
\[ \underset{a\leq t\leq b}{\text{ess sup}} [u^c|f(t)|]< \infty\qquad (p= \infty), \] for \(c\in\mathbb{R}= (-\infty,\infty)\), in particular in the space \(L^p(0,\infty)\) \((1\leq p\leq\infty)\). The existence almost everywhere is established for the corresponding Hadamard-type fractional derivative for a function \(g(x)\) such that \(x^\mu g(x)\) have \(\delta\) derivatives up to order \(n-1\) on \([a,b]\) and \(\delta^{n-1}[x^\mu g(x)]\) is absolutely continuous on \([a,b]\). Semigroup and reciprocal properties for the above operators are proved.

MSC:

26A33 Fractional derivatives and integrals
47B38 Linear operators on function spaces (general)
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