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Sharp integral inequalities for the dyadic maximal operator and applications

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Abstract

We prove a sharp integral inequality for the dyadic maximal function of \(\phi \in L^p\). This inequality connects certain quantities related to integrals of \(\phi \) and the dyadic maximal function of \(\phi \), under the hypothesis that the variables \(\int _X\phi \, \mathrm {d}\mu =f,\)\(\int _X\phi ^q\, \mathrm {d}\mu =A,\)\(1<q<p,\) are given, where \(0<f^q \le A.\) Additionally, it contains a parameter \(\beta >0\) 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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Authors and Affiliations

  1. Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, Zografou, 157 84, Athens, Greece

    Anastasios D. Delis

  2. Department of Mathematics, University of Ioannina, Panepistimioupolis, 451 10, Ioannina, Greece

    Eleftherios N. Nikolidakis

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  1. Anastasios D. Delis
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  2. Eleftherios N. Nikolidakis
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Correspondence to Eleftherios N. Nikolidakis.

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Delis, A.D., Nikolidakis, E.N. Sharp integral inequalities for the dyadic maximal operator and applications. Math. Z. 291, 1197–1209 (2019). https://doi.org/10.1007/s00209-019-02254-4

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  • DOI: https://doi.org/10.1007/s00209-019-02254-4

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