Fractional integrals and differentials of variable order in Hölder spaces $H^{\omega(t,x)}$

We consider generalized Hölder spaces of functions on the segment of real axis, whose local continuity modulus has a dominant which may vary from a point to point. We establish theorems on the mapping properties of fractional integrals of variable order, from such a variable generalized Hölder space to another one with a “better” dominant, and similar mapping properties of fractional differentials of variable order from such a space into the space with “worse” dominant. Variable order can take values between zero and unity.