Towards a Theory of Dobrakov-Sobolev Spaces

Abstract

The aim of this paper is to introduce a generalization of Sobolev spaces based on the Dobrakov integral. More precisely, we consider the setting of Banach-valued functions and Fomin differentiable Borel operator-valued measures on a finite-dimensional space. To build the necessary rigorous foundation, we establish analogs of several key results from the theory of differentiable real-valued measures, including the Leibniz rule and the integration by parts formula, all within the context of Dobrakov integration. These results are then embedded into the general scheme of vector-valued distribution theory. In particular, we describe the configuration of test spaces that yields an appropriate definition of a generalized derivative with respect to a differentiable operator-valued measure.