Hadamard–Bergman Operators on Weighted Spaces

This article continues the study of the Hadamard–Bergman operators in the unit disc of the complex plane. These operators arose as a natural generalization of the orthogonal projector and represent an integral realization of multiplier operators. Here we consider mainly the further development of the theory of such operators in the context of operators of variable order, that is, with kernels depending on an external variable. We give some important examples illustrating the relevance of such research, including the integrodifferentiation operator of variable order and certain Volterra type operator studied in complex analysis earlier. It is worth noting that it is the Hadamard–Bergman operators that seem to lay claim to the most direct complex analogue of operators with homogeneous kernels in the unit disc. This opens up excellent opportunities for further development of the study of operators of the Hadamard–Bergman type.