Weyl fractional integrals in the generalized Hölder classes (Q1382893)

For the Weyl fractional integration operator \[ J_{(\alpha )} \varphi =\frac {1}{2\pi} \int_0^{2 \pi} \varphi (x-t) \psi_\alpha (t) dt, \qquad \psi_\alpha (t)=\frac 1{\pi} \sum_{k=0}^{\infty} \frac{\cos \left(kt- \frac {\alpha \pi}{2}\right)}{k^\alpha},\quad \alpha >0, \] the authors show that \(J_{(\alpha )} \) realises an isomorphism between the generalised Hölder spaces \[ H_p^{\omega, m, k}=\{f: f \in L_p (0, 2\pi),\;\| \Delta_h^k f \| _p \leq M h^{-m} \omega (h) \} \] and \(H_p^{\omega_\alpha, m, k}\), \(p \geq 1\). Here \(k \in N\), \(m \geq 0\), \(\omega_\alpha=t^\alpha \omega (t)\) and \(\omega (t)\) satisfies some Bari-Stechkin type conditions. The investigation is based on the Zygmund type estimates for integral continuity modulus. The paper by the reviewer and \textit{Z. Mussalaeva} [Georgian Math. J. 1, No. 5, 537-559 (1994; Zbl 0810.26005)] is also relevent, in which similar problems were studied for the Riemann-Liouville fractional integration in the case \(p=\infty\), \(0< \alpha < 1\), but with a weight function.