Fractional integrals of imaginary order in the Hölder space on a segment (Q1594301)

The paper deals with the investigation of the fractional integrals in Hölder spaces on a finite interval of the real axis. It is known the classical result by Hardy and Littlewood that the fractional integral \(I^{\alpha}_{0+}\) of positive order \(\alpha >0\) isomorphically maps the space \(H^{\lambda}_{0}[0,1]\) of Hölder functions on \([0,1]\) vanishing at zero, onto the space \(H^{\lambda +\alpha}_{0}[0,1]\) [see Sections 3 and 13 in the book by \textit{S. G. Samko}, the reviewer and \textit{O. I. Marichev}, ``Fractional integrals and derivatives: Theory and applications'' (1993; Zbl 0818.26003; Russian original 1987; Zbl 0617.26004)]. The paper is devoted to prove such a statement for the fractional integral \(I^{\alpha}_{0+}\) with complex \(\alpha\). It is established that \[ I^{\alpha}_{0+}(H^{\lambda}_{0}[0,1])= H^{\lambda +\text{Re}(\alpha)}_{0}[0,1]\tag{1} \] provided \(0<\lambda <1\) and \(0<\lambda + \text{Re}(\alpha)<1\). The proof of this result is based on the properties (such as strong continuity, analogues of Zygmund's estimates and others) for the fractional integral \(I^{i\theta}_{0+}\) of purely imaginary order \(I^{i\theta}_{0+}\) in the Marchaud form \[ (I^{i\theta}_{0+}f)(x)={\frac{x^{i\theta}f(x)}{\Gamma (1+i\theta)}} +{\frac{i\theta}{\Gamma (1+i\theta)}} \int^{x}_{0}{\frac{f(x)-f(t)}{(x-t)^{1-i\theta}}} dt. \] The analogy of the isomorphic relation (1) is proved for the Hölder weighted space of functions on \([0,1]\) with a power weight concentrated at zero.