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Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1197–1209 | Cite as

Sharp integral inequalities for the dyadic maximal operator and applications

  • Anastasios D. Delis
  • Eleftherios N. NikolidakisEmail author
Article
  • 101 Downloads

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 fAq,  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.

Keywords

Bellman Dyadic maximal function Integral inequality 

Mathematical Subject Classification

42B25 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Anastasios D. Delis
    • 1
  • Eleftherios N. Nikolidakis
    • 2
    Email author
  1. 1.Department of MathematicsNational and Kapodistrian University of AthensAthensGreece
  2. 2.Department of MathematicsUniversity of IoanninaIoanninaGreece

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