(p, q)-Regular operators between Banach lattices

Pérez, Enrique Alfonso Sánchez; Tradacete, Pedro

doi:10.1007/s00605-018-1247-y

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…For ϕ = φ = Id R , we get the definition of a (p, q)-regular operator [20,21]; the case p = q is the most relevant from the point of view of the classical theory [2].…”

Section: Separation Arguments For Lipschitz Pointwise Transformations...mentioning

confidence: 99%

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

Arnau,

Sánchez-Pérez

2023

Mathematics

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We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [−1,1]) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces.

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“…For ϕ = φ = Id R , we get the definition of a (p, q)-regular operator [20,21]; the case p = q is the most relevant from the point of view of the classical theory [2].…”

Section: Separation Arguments For Lipschitz Pointwise Transformations...mentioning

confidence: 99%

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

Arnau,

Sánchez-Pérez

2023

Mathematics

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“…We refer the reader to [31] for a more recent account on this and the closely related notions of (p, q)-regularity.…”

Section: Regular Operators On Subspaces Of L Pmentioning

confidence: 99%

Strictly singular non-compact operators between $L_p$ spaces

Hernández¹,

Semenov²,

Tradacete³

2020

Preprint

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We study the structure of strictly singular non-compact operators between L p spaces. Answering a question raised in [17], it is shown that there exist operators T , for which the set of points ( 1p , 1 q ) ∈ (0, 1) × (0, 1) such that T : L p → L q is strictly singular but not compact contains a line segment in the triangle {( 1p , 1 q ) : 1 < p < q < ∞} of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular operators defined on subspaces of L p is also explored. i p,q : ℓ p ֒→ ℓ q 2010 Mathematics Subject Classification. 47B07, 46E30, 28A78.

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“…those which can be written as a difference of two positive operators-, was first considered in [3] in connection with the interpolation theory of Banach lattices (and also implicitly in [19]). In particular, these operators are related to the so-called Marcinkiewicz-Zygmund inequalities [7] and other interpolation properties (see [25,27] for recent developments about this).…”

Section: Introductionmentioning

confidence: 99%

$p$-Regularity and Weights for Operators Between $L^p$-Spaces

Pérez

Tradacete

2020

Z. Anal. Anwend.

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We explore the connection between p-regular operators on Banach function spaces and weighted p-estimates. In particular, our results focus on the following problem. Given finite measure spaces µ and ν, let T be an operator defined from a Banach function space X(ν) and taking values on L p (vdµ) for v in certain family of weights V ⊂ L 1 (µ) + : we analyze the existence of a bounded family of weights W ⊂ L 1 (ν) + such that for every v ∈ V there is w ∈ W in such a way that T : L p (wdν) → L p (vdµ) is continuous uniformly on V . A condition for the existence of such a family is given in terms of p-regularity of the integration map associated to a certain vector measure induced by the operator T .2010 Mathematics Subject Classification. 46E30,46B42,47B10. Key words and phrases. Banach function space; p-regular operator; weighted p-estimate. E. A. Sánchez Pérez gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grant MTM2016-77054-C2-1-P. P. Tradacete gratefully acknowledges support of Spanish Ministerio de Economía, Industria y Competitividad through grants MTM2016-76808-P and MTM2016-75196-P. PreliminariesLet (Ω, Σ, µ) be a finite measure space. Let L 0 (µ) be the space of all measurable real functions on Ω, identifying functions that are equal µ-a.e. We say that a Banach function space over µ is a Banach space X(µ) ⊆ L 0 (µ) that contains all simple functions, and whenever |f | ≤ |g| and g ∈ X(µ), then f ∈ X(µ) with f X(µ) ≤ g X(µ) . For the aim of simplicity, we will write sometimes X instead of X(µ) if the measure does not play a relevant role or is clear in the context. Recall X(µ) is order continuous if for every sequence (f n ) n ∈ X(µ), f n ↓ 0 implies f n X(µ) → 0. In this case, the Köthe dual X(µ) ′ = {g ∈ L 0 (µ) : f g ∈ L 1 (µ) whenever f ∈ X(µ)}, p-REGULAR OPERATORS AND WEIGHTS

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(p, q)-Regular operators between Banach lattices

Cited by 4 publications

References 17 publications

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

Lipschitz Transformations and Maurey-Type Non-Homogeneous Integral Inequalities for Operators on Banach Function Spaces

Strictly singular non-compact operators between $L_p$ spaces

$p$-Regularity and Weights for Operators Between $L^p$-Spaces

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