On the Theory of Spaces of Generalized Bessel Potentials

We define the weighted Dirichlet integral and show that this integral can be represented by a multidimensional generalized shift. The corresponding norm does not allow us to define the function spaces of arbitrary fractional order of smoothness, and so we introduce the new norm that is related to a generalized Bessel potential. Potential theory originates from the theory of electrostatic and gravitational potentials and the study of the Laplace, wave, Helmholtz, and Poisson equations. The celebrated Riesz potentials are the realizations of the real negative powers of the Laplace and wave operators. In the meantime, much attention in potential theory is paid to the Bessel potential generating the spaces of fractional smoothness. We progress in generalization by considering the Laplace–Bessel operator constructed from the singular Bessel differential operator. The theory of singular differential equations with the Bessel operator as well as the theory of the corresponding weighted function spaces are closely connected and belong to the areas of mathematics whose theoretical and applied significance can hardly be overestimated.