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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
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.
Received: 24.12.2013
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
Linking options:
https://www.mathnet.ru/eng/rm9568https://doi.org/10.1070/RM2014v069n04ABEH004908 https://www.mathnet.ru/eng/rm/v69/i4/p103
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Abstract page: | 636 | Russian version PDF: | 217 | English version PDF: | 27 | References: | 74 | First page: | 38 |
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