Optimal weak type estimates for dyadic-like maximal operators

Nikolidakis, Eleftherios N.

doi:10.5186/aasfm.2013.3817

Cited by 12 publications

(13 citation statements)

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“…for every φ ∈ L 1 (R n ) and every λ > 0, and which is easily proved to be best possible. Further refinements of (1.2) can be seen in [9] and [10]. Then by using (1.2) and the well known Doob's method it is not difficult to prove that the following L p inequality is also true for every p > 1 and φ ∈ L p (R n ).…”

Section: Introductionmentioning

confidence: 99%

Sharp integral inequalities for the dyadic maximal operator and applications

Delis

Nikolidakis

2019

Math. Z.

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We prove a sharp integral inequality for the dyadic maximal function of φ ∈ L p . This inequality connects certain quantities related to integrals of φ and the dyadic maximal function of φ, under the hypothesis that the variables X φ dµ = f, X φ q dµ = A, 1 < q < p, are given, where 0 < f q ≤ A. Additionally, it contains a parameter β > 0 which when it attains a certain value depending only on f, A, q, the inequality becomes sharp. Using this inequality we give an alternative proof of the evaluation of the Bellman function related to the dyadic maximal operator of two integral variables.

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Section: Introductionmentioning

confidence: 99%

Sharp integral inequalities for the dyadic maximal operator and applications

Delis

Nikolidakis

2019

Math. Z.

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“…For the study of the dyadic maximal operator it is desirable for one to find refinements of the above mentioned inequalities. Concerning (1.2), improvements have been given in [9] and [8]. If we consider (1.3), there is a refinement of it if one fixes the L 1 -norm of φ.…”

Section: Introductionmentioning

confidence: 99%

A sharp integral inequality for the dyadic maximal operator and a related stability result

Nikolidakis¹

2020

Ann. Acad. Sci. Fenn. Math.

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We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in [3]. Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions, under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.https://doi.org/10.5186/aasfm.2020.4517 2010 Mathematics Subject Classification: Primary 42B25.

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“…Our aim in this article is to study this maximal operator and one way to do this is to find certain refinements of the inequalities satisfied by it such as (1.2) and (1.3). Concerning (1.2) refinements have been made in [8], [10] and [12]. Refinements of (1.3) can be found in [5] or even more general in [6].…”

Section: Introductionmentioning

confidence: 99%