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Fractional integration operator of variable order in the Hölder spaces ${H}^{\lambda \left(x\right)}$. (English) Zbl 0838.26005

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

${I}_{a+}^{\alpha \left(x\right)}\varphi =\frac{1}{{\Gamma }\left(\alpha \left(x\right)\right)}{\int }_{0}^{x}\varphi \left(t\right){\left(x-t\right)}^{\alpha \left(x\right)-1}dt\phantom{\rule{2.em}{0ex}}\left(-\infty

of variable order $\alpha \left(x\right)>0$ in generalized Hölder spaces ${H}^{\lambda \left(x\right)}\left[a,b\right]$, the order $\lambda \left(x\right)$ $\left(0<\lambda \left(x\right)\le 1\right)$ of which also depends on the point $x$. The main result is the theorem on the behavior of the operator ${I}_{a+}^{\alpha \left(x\right)}$ in the space ${H}^{\lambda \left(x\right)}\left[a,b\right]$. 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)].

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