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Russian Mathematical Surveys, 2014, Volume 69, Issue 4, Pages 681–741
DOI: https://doi.org/10.1070/RM2014v069n04ABEH004908
(Mi rm9568)
 

This article is cited in 5 scientific papers (total in 5 papers)

Interpolation functions and the Lions–Peetre interpolation construction

V. I. Ovchinnikov

Voronezh State University
References:
Abstract: The generalization of the Lions–Peetre interpolation method of means considered in the present survey is less general than the generalizations known since the 1970s. However, our level of generalization is sufficient to encompass spaces that are most natural from the point of view of applications, like the Lorentz spaces, Orlicz spaces, and their analogues. The spaces $\varphi(X_0,X_1)_{p_0,p_1}$ considered here have three parameters: two positive numerical parameters $p_0$ and $p_1$ of equal standing, and a function parameter $\varphi$. For $p_0\ne p_1$ these spaces can be regarded as analogues of Orlicz spaces under the real interpolation method. Embedding criteria are established for the family of spaces $\varphi(X_0,X_1)_{p_0,p_1}$, together with optimal interpolation theorems that refine all the known interpolation theorems for operators acting on couples of weighted spaces $L_p$ and that extend these theorems beyond scales of spaces. The main specific feature is that the function parameter $\varphi$ can be an arbitrary natural functional parameter in the interpolation.
Bibliography: 43 titles.
Keywords: interpolation spaces, interpolation functors with function parameters, interpolation orbits, orbits with respect to von Neumann–Schatten operators, optimal interpolation theorems, embedding theorems for Orlicz–Sobolev spaces.
Funding agency Grant number
Russian Foundation for Basic Research 13-01-00378
This research was supported by the Russian Foundation for Basic Research (grant no. 13-01-00378).
Received: 24.12.2013
Russian version:
Uspekhi Matematicheskikh Nauk, 2014, Volume 69, Issue 4(418), Pages 103–168
DOI: https://doi.org/10.4213/rm9568
Bibliographic databases:
Document Type: Article
UDC: 517.982
MSC: Primary 46B70; Secondary 46M35, 47A57
Language: English
Original paper language: Russian
Citation: V. I. Ovchinnikov, “Interpolation functions and the Lions–Peetre interpolation construction”, Uspekhi Mat. Nauk, 69:4(418) (2014), 103–168; Russian Math. Surveys, 69:4 (2014), 681–741
Citation in format AMSBIB
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  • https://doi.org/10.1070/RM2014v069n04ABEH004908
  • https://www.mathnet.ru/eng/rm/v69/i4/p103
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Успехи математических наук Russian Mathematical Surveys
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    Abstract page:636
    Russian version PDF:217
    English version PDF:27
    References:74
    First page:38
     
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