Fractional Integro-Differentiation of the Complex Order in Generalized Holder Spaces

Zygmund type estimates are proved for the fractional integro-differentiation operators I_+^\alpha and D^\alpha _+ = I_+^{-\alpha} of the complex order \alpha, 0 \le \Re \alpha\lt 1 . Applications are given to the operator I_+^\alpha to be an isomorphism between generalized Holder spaces H^\omega _0([0,1],\rho) and H^{\omega_\alpha}_0([0,1],\rho), where \rho is a power weight and \omega_\alpha (t)=t^{\Re \alpha} \omega (t) .

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