Some remarks on dimension-free estimates for the discrete Hardy—Littlewood maximal functions

Kosz, Dariusz; Mirek, Mariusz; Plewa, Paweł; Wróbel, Błażej

doi:10.1007/s11856-022-2382-7

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…Consequently, , which implies . By using Proposition 2.1 and the fact that (see [16] and [13, Theorem 1]), we conclude that .…”

Section: Proof Of Theorem 19mentioning

confidence: 67%

“…The inequality follows by repeating the arguments used to prove [13, equation (1.3)]. We only sketch the proof.…”

Section: Introductionmentioning

confidence: 95%

“…For the centered operator, the situation is different and only the lower bound for can be easily transferred from the continuous setting to the discrete one, cf. [13, Theorem 1].…”

Section: Introductionmentioning

confidence: 99%

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Sharp constants in inequalities admitting the Calderón transference principle

Kosz

2023

Ergod. Th. Dynam. Sys.

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The aim of this note is twofold. First, we prove an abstract version of the Calderón transference principle for inequalities of admissible type in the general commutative multilinear and multiparameter setting. Such an operation does not increase the constants in the transferred inequalities. Second, we use the last information to study a certain dichotomy arising in problems of finding the best constants in the weak type $(1,1)$ and strong type $(p,p)$ inequalities for one-parameter ergodic maximal operators.

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“…Consequently, , which implies . By using Proposition 2.1 and the fact that (see [16] and [13, Theorem 1]), we conclude that .…”

Section: Proof Of Theorem 19mentioning

confidence: 67%

“…The inequality follows by repeating the arguments used to prove [13, equation (1.3)]. We only sketch the proof.…”

Section: Introductionmentioning

confidence: 95%

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Sharp constants in inequalities admitting the Calderón transference principle

Kosz

2023

Ergod. Th. Dynam. Sys.

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On the solution of Waring problem with a multiplicative error term: Dimension-free estimates

Mirek¹,

Słomian

Wróbel³

2023

Proc. Amer. Math. Soc.

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Dimension-Free Estimates for the Discrete Spherical Maximal Functions

Mirek

Szarek

Wróbel

2023

International Mathematics Research Notices

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We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein, and Wainger) corresponding to the Euclidean spheres in $\mathbb {Z}^{d}$ with dyadic radii have $\ell ^{p}(\mathbb {Z}^{d})$ bounds for all $p\in [2, \infty ]$ independent of the dimensions $d\ge 5$. An important part of our argument is the asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term. By considering new approximating multipliers, we will show how to absorb an exponential in dimension (like $C^{d}$ for some $C>1$) growth in norms arising from the sampling principle of Magyar, Stein, and Wainger and ultimately deduce dimension-free estimates for the discrete spherical maximal functions.

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Some remarks on dimension-free estimates for the discrete Hardy—Littlewood maximal functions

Cited by 3 publications

References 15 publications

Sharp constants in inequalities admitting the Calderón transference principle

Sharp constants in inequalities admitting the Calderón transference principle

On the solution of Waring problem with a multiplicative error term: Dimension-free estimates

Dimension-Free Estimates for the Discrete Spherical Maximal Functions

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