title: On the generalized Bessel potential and perfect functional completion
metadata.name: dc.contributor.author
metadata.value: Dzhabrailov, Аkhmed L.
metadata.lang: -
metadata.name: dc.contributor.author
metadata.value: Shishkina, Elina L.
metadata.lang: -
metadata.name: dc.date.accessioned
metadata.value: 2023-05-18T21:34:44Z
metadata.lang: -
metadata.name: dc.date.available
metadata.value: 2023-05-18T21:34:44Z
metadata.lang: -
metadata.name: dc.date.issued
metadata.value: 2023-05
metadata.lang: -
metadata.name: dc.identifier.citation
metadata.value: Dzhabrailov А. L., Shishkina E. L. On the generalized Bessel potential and perfect functional completion. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 2023, vol. 10 (68), issue 2, pp. 200–211. https://doi.org/10.21638/spbu01.2023.202 (In Russian)
metadata.lang: en_GB
metadata.name: dc.identifier.other
metadata.value: https://doi.org/10.21638/spbu01.2023.202
metadata.lang: -
metadata.name: dc.identifier.uri
metadata.value: http://hdl.handle.net/11701/41478
metadata.lang: -
metadata.name: dc.description.abstract
metadata.value: 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.
metadata.lang: en_GB
metadata.name: dc.description.sponsorship
metadata.value: The work of A. L. Dzhabrailov was supported by the Ministry of Education and Science of the
  Russian Federation under the state task FEGS-2020-0001.
metadata.lang: en_GB
metadata.name: dc.language.iso
metadata.value: ru
metadata.lang: en_GB
metadata.name: dc.publisher
metadata.value: St Petersburg State University
metadata.lang: en_GB
metadata.name: dc.relation.ispartofseries
metadata.value: Vestnik of St Petersburg University. Mathematics. Mechanics. Astronomy;Volume 10; Issue 2
metadata.lang: -
metadata.name: dc.subject
metadata.value: generalized Bessel potential
metadata.lang: en_GB
metadata.name: dc.subject
metadata.value: perfect completion of spaces
metadata.lang: en_GB
metadata.name: dc.title
metadata.value: On the generalized Bessel potential and perfect functional completion
metadata.lang: en_GB
metadata.name: dc.type
metadata.value: Article
metadata.lang: en_GB
resources.filename: 200-211.pdf
resources.url: https://dspace.spbu.ru/bitstream/11701/41478/1/200-211.pdf
resources.format: Adobe PDF
resources.size: 352,43 kB