On Generalized Bessel Potentials and Perfect Functional Completions

Abstract

The class of generalized Bessel potentials is the main object of study in this paper. The generalized Bessel potential is a negative real power of the operator (I – ∆_γ), where ∆_γ = $\sum\nolimits_{k = 1}^n {\frac{1}{{x_{k}^{{{{\gamma }_{k}}}}}}\frac{\partial }{{\partial {{x}_{k}}}}x_{k}^{{{{\gamma }_{k}}}}\frac{\partial }{{\partial {{x}_{k}}}}}$ is the Laplace–Bessel operator and γ = (γ₁, …, γ_n) is a multi-index consisting of positive fixed real numbers. To solve various problems for differential equations, prove embedding theorems for some classes of functions, and invert integral operators, there is a need to consider functions up to some small (from the point of view of the problem under consideration) set. As such a small set, a set of Lebesgue measure zero is often taken. However, for many problems, sets of Lebesgue measure zero turn out to be too large to be disregarded. For example, when a boundary problem is solved, the behavior of the solution at the boundary is essential. In this regard, there arose the need to construct complete classes of admissible functions suitable for solving specific problems. Two stages of constructing a functional completion were presented by N. Aronszajn and K.T. Smith. The first of these stages consists in finding a suitable class of exceptional sets. The second is to find functions defined modulo of these exceptional classes that need to be joined 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 in fact to one functional completion. It is clear that the most suitable functional completion is the one whose exceptional class is the smallest, since the functions will then 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 using the norm associated with the kernel of the generalized Bessel potential.