A note on potentials of imaginary order in holomorphic Hölder spaces

In this note we study a generalization of a holomorphic Bergman-type projection, the operator of fractional integro-differentiation of imaginary order, in generalized variable Holder spaces. We consider two definitions of variable generalized Holder spaces of holomorphic functions: one defined directly via variable modulus of continuity, while the other is defined in terms of behavior of derivatives near the boundary. We prove boundedness results for the operator of fractional integrodifferentiation of imaginary order in the scales of these spaces.