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Mat. Zametki, 2018, Volume 104, Issue 3, Pages 467–480 (Mi mz12118)  

This article is cited in 1 scientific paper (total in 1 paper)

Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on $\mathbb{R}^n$

S. M. Umarkhadzhievab

a Academy of Sciences of Chechen Republic
b Complex Research Institute named after Kh. I. Ibragimov, Russian Academy of Sciences, Groznyi

Abstract: The Riesz potentials $I^\alpha f$, $0<\alpha<\infty$, are considered in the framework of a grand Lebesgue space $L^{p),\theta}_a$, $1<p<\infty$, $\theta>0$, on $\mathbb{R}^n$ with grandizers $a\in L^1(\mathbb{R}^n)$, which are understood in the case $\alpha\ge n/p$ in terms of distributions on test functions in the Lizorkin space. The images under $I^\alpha$ of functions in a subspace of the grand space which satisfy the so-called vanishing condition is studied. Under certain assumptions on the grandizer, this image is described in terms of the convergence of truncated hypersingular integrals of order $\alpha$ in this subspace.

Keywords: Riesz potential, space of Riesz potentials, hypersingular integral, grand Lebesgue space, grandizer, Lizorkin space of test functions, identity approximation.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00094-A
17-301-50023-мол-нр
This work was supported by the Russian Foundation for Basic Research under grants 17-301-50023-mol-nr and 18-01-00094-A.


DOI: https://doi.org/10.4213/mzm12118

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English version:
Mathematical Notes, 2018, 104:3, 454–464

Bibliographic databases:

UDC: 517.982+517.983
Received: 30.11.2017

Citation: S. M. Umarkhadzhiev, “Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on $\mathbb{R}^n$”, Mat. Zametki, 104:3 (2018), 467–480; Math. Notes, 104:3 (2018), 454–464

Citation in format AMSBIB
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\paper Description of the Space of Riesz Potentials of Functions in a Grand Lebesgue Space on~$\mathbb{R}^n$
\jour Mat. Zametki
\yr 2018
\vol 104
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\pages 467--480
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\pages 454--464
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. M. Umarkhadzhiev, “Ob ellipticheskikh odnorodnykh differentsialnykh operatorakh v grand-prostranstvakh”, Izv. vuzov. Matem., 2020, no. 3, 64–73  mathnet  crossref
  • Математические заметки Mathematical Notes
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