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.

中文翻译：

---

关于广义贝塞尔势和完美的函数完成

摘要

广义贝塞尔势类是本文的主要研究对象。广义贝塞尔势是算子的负实幂 ( I – ∆ _γ )，其中 ∆ _γ = \(\sum\nolimits_{k = 1}^n {\frac{1}{{x_{k}^{ {{{\gamma }_{k}}}}}}\frac{\partial }{{\partial {{x}_{k}}}}x_{k}^{{{{\gamma }_{ k}}}}\frac{\partial }{{\partial {{x}_{k}}}}} \) 是拉普拉斯-贝塞尔算子，γ = (γ ₁ , …, γ _n) 是由固定正实数组成的多重指标。为了解决微分方程的各种问题，证明某些类函数的嵌入定理，以及求逆积分运算符，需要将函数考虑到一些小的（从所考虑的问题的角度来看）集合。作为这样的小集，往往取勒贝格零测度集。然而，对于许多问题，Lebesgue 测度零集太大而无法忽略。例如，当解决边界问题时，解决方案在边界处的行为是必不可少的。在这方面，需要构建适用于解决特定问题的完整的可接受函数类。N. Aronszajn 和 KT Smith 介绍了构建功能完成的两个阶段。这些阶段中的第一个阶段在于找到合适的异常集类。第二个是找到需要连接以获得完整功能类的这些异常类的模定义函数。事实证明，在一个特定问题中可以有无限多个合适的异常类，但实际上它们中的每一个都对应一个功能完成。很明显，最合适的函数完成是异常类最小的函数完成，因为函数将以尽可能高的准确性定义。只要存在这样的最小异常类，相应的功能完成就称为完美完成。在本文中，完美补全是使用与广义贝塞尔势核相关的范数构造的。