Operators with homogeneous kernels

Rutitskii, Ya. B.

doi:10.1007/bf00970129

Cited by 3 publications

(5 citation statements)

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“…These operators are also bounded in some Banach ideal spaces which are not symmetric. For example, if X = L p (x α ), then σ s X→X = s 1/p+α (see [Ru80] for more examples). The Boyd indices of a symmetric space X are defined by For every m ∈ N let σ m and σ 1/m be the dilation operators defined in spaces of sequences a = (a n ) by: If an increasing concave function ϕ is defined on [0, 1] (resp.…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

Isomorphic Structure of Cesàro and Tandori Spaces

Асташкин

Leśnik²,

Maligrańda³

2019

Can. J. Math.-J. Can. Math.

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Abstract. We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. e main result states that the Cesàro function space Ces ∞ and its sequence counterpart ces ∞ are isomorphic, which answers the question posted in [?].is is rather surprising since Ces ∞ (like the known Talagrand's example [?]) has no natural lattice predual. We prove that ces ∞ is not isomorphic to ℓ ∞ nor is Ces ∞ isomorphic to the Tandori spaceL with the norm f L = f L , wherẽ f (t) ∶= ess sup s≥t f (s) . Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro-Marcinkiewicz and Cesàro-Lorentz spaces have the latter property.

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Section: Definitions and Basic Factsmentioning

confidence: 99%

Isomorphic Structure of Cesàro and Tandori Spaces

Асташкин

Leśnik²,

Maligrańda³

2019

Can. J. Math.-J. Can. Math.

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“…These operators are also bounded in some Banach function spaces which are not necessary symmetric. For example, if either X = L p (x α ) or X = C(L p (x α )), then σ τ X→X = τ 1/p+α (see [Ru80] for more examples).…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

“…Properties of Cesàro sequence spaces ces p = Cl p were investigated in many papers (see [MPS07] and references given there), while properties of Cesàro function spaces Ces p (I) = CL p (I) we can find in [AM09] and [AM14b]. Abstract Cesàro spaces CX for Banach ideal spaces X on [0, ∞) were defined already in [Ru80] and spaces CX, X for X being a symmetric space on [0, ∞) have appeared, for example, in [KMS07], [DS07] and [AM13]. General considerations of abstract Cesàro spaces began to be studied in papers [LM15a,LM15b].…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

Interpolation of abstract Cesàro, Copson and Tandori spaces

Leśnik

Maligrańda

2016

Indagationes Mathematicae

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We study real and complex interpolation of abstract Cesàro, Copson and Tandori spaces, including the description of Calderón-Lozanovskiǐ construction for those spaces. The results may be regarded as generalizations of interpolation for Cesàro spaces Ces p (I) in the case of real method, but they are new even for Ces p (I) in the case of complex method. Some results for more general interpolation functors are also presented. The investigations show an interesting phenomenon that there is a big difference between interpolation of Cesàro function spaces in the cases of finite and infinite interval.

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“…The space CX for a Banach ideal space X on [0, ∞) was defined already in [Ru80] and spaces CX, X for X being a symmetric space on [0, ∞) have appeared, for example, in [KMS07] and [DS07].…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

“…They are also bounded in some Banach ideal spaces which are not necessary symmetric. For example, if either X = L p (x α )(I) or X = CL p (x α )(I), then σ τ X→X = τ 1/p+α (see [Ru80] for more examples).…”

Section: Definitions and Basic Factsmentioning

confidence: 99%

Abstract Cesàro Spaces. I. Duality

Leśnik,

Maligranda

2014

Preprint

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We study abstract Cesàro spaces CX, which may be regarded as generalizations of Cesàro sequence spaces ces p and Cesàro function spaces Ces p (I) on I = [0, 1] or I = [0, ∞), and also as the description of optimal domain from which Cesàro operator acts to X. We find the dual of such spaces in a very general situation. What is however even more important, we do it in the simplest possible way. Our proofs are more elementary than the known ones for ces p and Ces p (I). This is the point how our paper should be seen, i.e. not as generalization of known results, but rather like grasping and exhibiting the general nature of the problem, which is not so easy visible in the previous publications. Our results show also an interesting phenomenon that there is a big difference between duality in the cases of finite and infinite interval. *This publication has been produced during scholarship period of the first author at the Luleå University of Technology, thanks to a Swedish Institute scholarschip (number 0095/2013).

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Operators with homogeneous kernels

Cited by 3 publications

References 2 publications

Isomorphic Structure of Cesàro and Tandori Spaces

Isomorphic Structure of Cesàro and Tandori Spaces

Interpolation of abstract Cesàro, Copson and Tandori spaces

Abstract Cesàro Spaces. I. Duality

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