Boundedness of classical operators in weighted spaces of holomorphic functions

We establish some criteria of the boundedness for some classical operators acting from an abstract Banach space of holomorphic functions in a complex domain to a weighted space of the same functions equipped with sup-norm. It is presented a further development of Zorboska’s idea that conditions of the boundedness of weighted composition operators including multiplication and usual composition ones and Volterra operator can be formulated in terms of $\delta$-functions norms in the corresponding dual spaces. As a consequence we obtain criteria of the boundedness of the above mentioned operators on generalized Bergman and Fock spaces. In particular cases it is possible to state these criteria in terms of weights defining spaces and functions giving the composition operator. In comparison with the previous results we essentially extend the class of weighted holomorphic spaces in the unit disc that admits a realization of Zorboska’s method. In addition, we develop an extension of this approach to weighted spaces of entire functions. In this relation we introduce the class of almost harmonic weights and obtain some estimates of $\delta$-functions norms in spaces dual to the generalized Fock spaces giving by almost harmonic weights.