Sharp integral inequalities for the dyadic maximal operator and applications

Abstract

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.

Sharp and general estimates for the Bellman function of three integral variables related to the dyadic maximal operator

“…Regarding the general case now, in Section 4 we provide an upper bound for (1.9) (Lemma 4.2). As a first step towards this, we prove an inequality satisfied by the corresponding maximal operator (Lemma 4.1) which can be interpreted as the basic inequality proved in [3], for the case β = 0.…”

Γεωμετρικά Προβλήματα Στη Θεωρία Διαφορισιμότητας

“…The first problem that concerned us was refining (1.5) even further. The results presented in [3] and [4] reflect these efforts. In particular, we also consider the q-norm, 1 < q < p, of the function ϕ as fixed and aim to compute the Bellman function of three integral variables which generalizes (1.40),…”

“…Για τη μελέτη του δυαδικού μεγιστικού τελεστή είναι προτιμότερο να περιοριστούμε σε συναρτήσεις που έχουν στήριγμα στο μοναδιαίο κύβο Το πρώτο πρόβλημα με το οποίο ασχοληθήκαμε κατά τη διάρκεια του διδακτορικού ήταν η περαιτέρω βελτίωση της (5.3). Τα αποτελέσματα που εμϕανίζονται στα [3] και [4]…”

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

“…For the case q = 1 and the value β = 1 p−1 inequality (1.10) is well known and is in fact equality, as can be seen by applying a simple integration by parts argument. We also note that inequality (1.8) is also a consequence of the results in [3], where it is proved a more general inequality which involves also the parameter A =´X φ q dµ. In this paper we ignore this parameter and give a more direct proof of (1.8).…”

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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.

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