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Fractional integration operator of variable order in the Hölder spaces \(H^{\lambda (x)}\). (English) Zbl 0838.26005
The paper deals with the mapping properties of the generalized Riemann-Liouville fractional integration operators \[ I_{a+}^{\alpha(x)} \varphi= {1\over {\Gamma (\alpha (x))}} \int^x_0 \varphi(t) (x-t)^{\alpha(x)-1} dt \qquad (-\infty< a<b< +\infty) \] of variable order \(\alpha (x)>0\) in generalized Hölder spaces \(H^{\lambda (x)} [a,b ]\), the order \(\lambda (x)\) \((0< \lambda(x)\leq 1)\) of which also depends on the point \(x\). The main result is the theorem on the behavior of the operator \(I_{a+}^{\alpha (x)}\) in the space \(H^{\lambda (x)} [a,b ]\). This statement generalizes the well-known Hardy-Littlewood theorem for the Riemann-Liouville fractional integrals in usual Hölder spaces [see Theorem 3.1 in the book by S. G. Samko, the reviewer and O. I. Marichev: “Integrals and derivatives of fractional order and some of their applications” (1987; Zbl 0617.26004; English translation 1993; Zbl 0818.26003)].
Reviewer: A.A.Kilbas (Minsk)

MSC:
26A33 Fractional derivatives and integrals
45P05 Integral operators
47B38 Linear operators on function spaces (general)
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