The class of convolution operators on the Marcinkiewicz spaces

Lau, Ka–Sing

doi:10.5802/aif.845

Cited by 5 publications

(5 citation statements)

References 4 publications

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“…where the limit is taken in the M -norm. The function in (12) is the convolution of and , and we write ( − ) ( ) . (15) In particular,…”

Section: Convolution On Marcinkiewicz Spacesmentioning

confidence: 99%

“…The quotient of M with respect to the null space I of the semi-norm is therefore a Banach space, which we denote bỹ M . Marcinkiewicz spaces have been studied or used in [3,7,[9][10][11][12][13][14]. In particular, it has been observed in [11] that all regular bounded Borel measures give rise to bounded convolution operators on M (R), with norm bounded by the norm of the measure, just as it happens for spaces.…”

Section: Introductionmentioning

confidence: 99%

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Function Spaces with BoundedLpMeans and Their Continuous Functionals

Picardello

2014

Abstract and Applied Analysis

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This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitableLpmeans. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.

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“…where the limit is taken in the M -norm. The function in (12) is the convolution of and , and we write ( − ) ( ) . (15) In particular,…”

Section: Convolution On Marcinkiewicz Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Function Spaces with BoundedLpMeans and Their Continuous Functionals

Picardello

2014

Abstract and Applied Analysis

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“…An interesting way to introduce mild decay conditions is based upon boundedness of integral averages over large intervals in R. This leads to Marcinkiewicz spaces M p (R) defined in Section 2. These spaces have been studied also in [3] and in [1].…”

Section: Introductionmentioning

confidence: 99%

“…In [3], K.S. Lau studies the convolution operators on M p , p ≥ 1, and on its closed subspace of regular functions, i.e.…”

Section: Introductionmentioning

confidence: 99%

Convolution on spaces of locally summable functions

Andreano

Grande

2008

Journal of Function Spaces

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“…Marcinkiewicz spaces M p (R) were introduced in [4] and studied in [3,5,[12][13][14] and [2]. M p (R) is defined as the space of functions f ∈ L p on all compact sets in R, 1 p < ∞, such that lim sup…”

Section: Introduction: Some Classical Spaces Of Bounded L P -Meansmentioning

confidence: 99%

Approximate identities on some homogeneous Banach spaces

Andreano

Picardello

2009

Monatsh Math

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We study convergence of approximate identities on some complete seminormed or normed spaces of locally L p functions where translations are isometries, namely Marcinkiewicz spaces M p and Stepanoff spaces S p , 1 p < ∞, as well as others where translations are not isometric but bounded (the bounded p-mean spaces M p ) or even unbounded (M p 0 ). We construct a function f that belongs to these spaces and has the property that all approximate identities φ ε * f converge to f pointwise but they never converge in norm.

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The class of convolution operators on the Marcinkiewicz spaces

Cited by 5 publications

References 4 publications

Function Spaces with BoundedLpMeans and Their Continuous Functionals

Function Spaces with BoundedLpMeans and Their Continuous Functionals

Convolution on spaces of locally summable functions

Approximate identities on some homogeneous Banach spaces

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