Quadratic interpolation and some operator inequalities

Fan, Mingzhi

doi:10.7153/jmi-05-36

Cited by 3 publications

(10 citation statements)

References 11 publications

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“…Complex interpolation with derivatives. In [15], pp. 421-422, Fan considers the more general complex interpolation method C θ(n) for the n:th derivative.…”

Section: Appendix: the Complex Methods Is Quadraticmentioning

confidence: 99%

“…The identity J ν θ = K ̺ θ can now be recognized as a sharp (isometric) Hilbert space version of the equivalence theorem of Peetre, which says that the standard K θ and J θ -methods give rise to equivalent norms on the category of Banach couples (see [7]). The problem of determining the pairs ̺, ν having the property that the K ̺ and J ν methods give equivalent norms was studied by Fan in [15], Section 3. 6.6.…”

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confidence: 99%

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Interpolation Between Hilbert Spaces

Ameur¹

2019

Trends in Mathematics

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This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we discuss the characterizations of all interpolation spaces [2] and of all quadratic interpolation spaces [13], and we give connections to other results in the area. Interpolation theoretic notions1.1. Interpolation norms. When X, Y are normed spaces, we use the symbol L(X; Y ) to denote the totality of bounded linear maps T : X → Y with the operator normWhen X = Y we simply write L(X). Consider a pair of Hilbert (or even Banach-) spaces H = (H 0 , H 1 ) which is regular in the sense that H 0 ∩ H 1 is dense in H 0 as well as in H 1 . We assume that the pair is compatible in the sense that H i ⊂ M for i = 0, 1 where M is some Hausdorff topological vector space.We define the K-functional ( 1 ) for the couple H by

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“…Complex interpolation with derivatives. In [15], pp. 421-422, Fan considers the more general complex interpolation method C θ(n) for the n:th derivative.…”

Section: Appendix: the Complex Methods Is Quadraticmentioning

confidence: 99%

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confidence: 99%

Interpolation Between Hilbert Spaces

Ameur¹

2019

Trends in Mathematics

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“…It induces the Hilbert scale which connects the Hilbert spaces being interpolated and admits a natural generalization to the case where a general enough function is taken instead of a number as an interpolation parameter. This generalization appeared in [14, p. 278] and then was studied in [2,9,12,33] and some other works; see also monograph [30] that discuss it and its applications. Considering the quadratic interpolation, we restrict ourselves to the case of separable complex Hilbert spaces and mainly follow [30, Section 1.1].…”

Section: Condition 22 the Polynomialsmentioning

confidence: 99%

Parabolic problems in generalized Sobolev spaces

Los¹,

Mikhailets²,

Murach³

2021

CPAA

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<p style='text-indent:20px;'>We consider a general inhomogeneous parabolic initial-boundary value problem for a <inline-formula><tex-math id="M1">\begin{document}$ 2b $\end{document}</tex-math></inline-formula>-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers <inline-formula><tex-math id="M2">\begin{document}$ s $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ s/(2b) $\end{document}</tex-math></inline-formula> and with a function <inline-formula><tex-math id="M4">\begin{document}$ \varphi:[1,\infty)\to(0,\infty) $\end{document}</tex-math></inline-formula> that varies slowly at infinity. The function parameter <inline-formula><tex-math id="M5">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.</p>

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“…These and some other properties of the extended Hilbert scale are considered in Section 2 of this paper; they are proved in Section 3. Note that the above interpolation and interpolational properties of Hilbert scales are studied in articles [4,5,16,18,19,35,36,38,47,59,63] (see also monographs [39, Chapter 1], [51,Section 1.1], and [68,Chapters 15 and 30]). Among them, of fundamental importance for our investigation is Ovchinnikov's result [59,Theorem 11.4.1] on an explicit description (with respect to equivalence of norms) of all Hilbert spaces that are interpolation ones between arbitrarily chosen compatible Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

“…It follows directly from Theorem 2.6 (namely, from the equalities in (4.13)) and the result byFan [18, formula (2.3)] that inequality (4.17) holds true with a certain number c > 0 written instead of c 2 ψ,ν √ 8. Note that Fan[18] denotes the interpolation space [H 0 , H 1 ] ψ by H χ where χ(t) ≡ ψ 2 ( √ t) and that ψ is pseudoconcave on (0, ∞) if and only if so is χ, with the dilation function χ(t) ≡ ψ 2 ( √ t). (As in Section 2, [H 0 , H 1 ] is a regular pair of separable complex Hilbert spaces.)…”

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confidence: 99%

An extended Hilbert scale and its applications

Mikhailets,

Murach,

Zinchenko

2021

Preprint

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We propose a new viewpoint on Hilbert scales extending them by means of all Hilbert spaces that are interpolation ones between spaces on the scale. We prove that this extension admits an explicit description with the help of OR-varying functions of the operator generating the scale. We also show that this extended Hilbert scale is obtained by the quadratic interpolation (with function parameter) between the above spaces and is closed with respect to the quadratic interpolation between Hilbert spaces. We give applications of the extended Hilbert scale to interpolational inequalities, generalized Sobolev spaces, and spectral expansions induced by abstract and elliptic operators.

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Quadratic interpolation and some operator inequalities

Cited by 3 publications

References 11 publications

Interpolation Between Hilbert Spaces

Interpolation Between Hilbert Spaces

Parabolic problems in generalized Sobolev spaces

An extended Hilbert scale and its applications

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