On the theory of spaces of generalized Bessel potentials

The purpose of the article is to introduce norms in the space of generalized Bessel potentials based on the weighted Dirichlet integrals. First, we define weighted Dirichlet integral and show that this integral can be represented using multidimensional generalised translation. Next, we demonstrate that this norm does not allow to define function spaces of arbitrary fractional order of smoothness. The potential theory originates from the theory of electrostatic and gravitational potentials and the Laplace, wave, Helmholtz, and Poisson equations. The famous Riesz potentials are known to be realizations of the real negative powers of the Laplace and wave operators. In the meantime, a lot of attention in the potential theory is given to the Bessel potential. Generalization in the article is achieved by considering the Laplace-Bessel operator which is constructed on the basis of the singular Bessel differential operator. The theory of singular differential equations containing the Bessel operator and the theory of the corresponding weighted function spaces belong to those mathematical areas, the theoretical and applied significance of which can hardly be overestimated.