On the generalized Bessel potential and perfect functional completion

Abstract

The class of generalized Bessel potentials is the main object of study in this article. The generalized Bessel potential is a negative real power of the operator (I −Δγ), where Δγ = n

k=1 1 x γk k ∂ ∂xk xγk k ∂ ∂xk — Laplace—Bessel operator, γ = (γ1, . . . , γn) — multi-index consisting of positive fixed real numbers. When solving various problems for differential equations, proving embedding theorems for some classes of functions, and inverting integral operators, there is a need to consider functions up to some small, from the point of view of the problem under consideration, set. Often a set of Lebesgue measure zero is taken as such a small set. However, often the sets of Lebesgue measure zero turn out to be too large to be neglected. For example, when solving a boundary problem, the behavior of the solution on the boundary is essential. In this regard, there was a need to construct complete classes of admissible functions suitable for solving specific problems. N. Aronshine and K. Smith presented two stages of constructing a functional completion. The first of these is finding a suitable class of exceptional sets. The second is to find the functions defined modulo these exceptional classes that need to be attached to get a complete functional class. It turns out that there can be infinitely many suitable exceptional classes in a particular problem, but each of them corresponds essentially to one functional completion. It is clear that the most suitable functional completion is the one whose exceptional class is the smallest, since then the functions will be defined with the best possible accuracy. Whenever such a minimal exceptional class exists, the corresponding functional completion is called a perfect completion. In this paper, perfect completions are constructed from the norm associated with the kernel of the generalized Bessel potential.