Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets

Deleaval, Luc; Guédon, Olivier; Maurey, Bernard

doi:10.5802/afst.1567

Cited by 23 publications

(12 citation statements)

References 71 publications

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“…Hence it is relatively easy to verify in the noncommutative setting. This approach is partially inspired by the works of Bourgain and Carbery [Bou86a,Bou86b,Bou87,Car86,DGM18] on the dimension free bounds of Hardy-Littlewood maximal operators. We refer to Section 2 for the notions of noncommutative L p -spaces L p (V N (Γ)) and noncommutative maximal norms sup + n x n p .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Pointwise convergence of noncommutative Fourier series

Hong,

Wang,

Wang

2019

Preprint

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This paper is devoted to the study of convergence of Fourier series for nonabelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. Based on this framework, this work studies the refined counterpart of pointwise convergence of these Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions tending to 1 pointwise, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence for all 1 < p < ∞. In a similar fashion, we also obtain results for a large subclass of groups (as well as discrete quantum groups) with the Haagerup property and weak amenability. We also consider the analogues of Fejér means and Bochner-Riesz means in the noncommutative setting. Our results in particular apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain as a byproduct the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities associated with convex bodies. As an ingredient, our proof also provides a refined version of Junge-Le Merdy-Xu's square function estimates Hp(M) ≃ L • p (M) when p → 1.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Pointwise convergence of noncommutative Fourier series

Hong,

Wang,

Wang

2019

Preprint

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“…Note however that this bound might considerably overestimate the actual ratio, since in general interpolation will not yield the best possible constants, and occasionally it may yield bounds that are very far from optimality; for instance, it is known that for p = 2, for Lebesgue measure in R d , and for balls defined by an arbitrary norm, optimal constants are uniformly bounded by 140 in every dimension (see [DeGuMa,Theorem 5.2]). For cubes (balls with respect to the ℓ ∞ norm), J. Bourgain has proved that dimension independent bounds hold for every p > 1, cf.…”

Section: Definitions and General Resultsmentioning

confidence: 99%

Kissing numbers and the centered maximal operator

Aldaz

2018

Preprint

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We prove that in a metric measure space X, if for some p ∈ (1, ∞) there are uniform bounds (independent of the measure) for the weak type (p, p) of the centered maximal operator, then X satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type (1, 1) of the centered operator.In R d with any norm, the constants coming from the Besicovitch intersection property are bounded above by the translative kissing numbers. This leads to improved estimates on the uniform bounds satisfied by the centered maximal operator defined using euclidean balls, as well as the sharp constants in dimensions 2 and 3. For the centered maximal operator defined using ℓ ∞ -balls (cubes) we obtain the sharp uniform bounds 2 d .

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“…However, the dimension-free phenomenon in harmonic analysis had been apparent much earlier, see for instance [36,Chapter 14, §3 in Vol.II], as well as [32] and the references given there. We refer also to more recent results [1,2,8,9,10,11,18,23,30] and the survey article [16] for a very careful and detailed exposition of the subject.…”

Section: A Review Of the Current State Of The Artmentioning

confidence: 99%