Composition operators in generalized Hölder spaces

A new characterization of boundedness of the composition operators in generalized Hölder spaces is obtained in terms that do not use the derivative of the composition operator, but using some averaging construction which represents certain integration of the modulus of continuity involving the symbol of the composition operator. This approach also allows to recover previously known results for the standard weights $\omega(t)=t^\alpha$ with $0<\alpha<1$. Certain further results on characterization of the same spaces are obtained as well.