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Fractional integration operator of variable order in the Hölder spaces H λ(x) . (English) Zbl 0838.26005

The paper deals with the mapping properties of the generalized Riemann-Liouville fractional integration operators

I a+ α(x) φ=1 Γ(α(x)) 0 x φ(t)(x-t) α(x)-1 dt(-<a<b<+)

of variable order α(x)>0 in generalized Hölder spaces H λ(x) [a,b], the order λ(x) (0<λ(x)1) of which also depends on the point x. The main result is the theorem on the behavior of the operator I a+ α(x) in the space H λ(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)].

26A33Fractional derivatives and integrals (real functions)
45P05Integral operators
47B38Operators on function spaces (general)