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An eigenfunction stability estimate for approximate extremals of the Bellman function for the dyadic maximal operator on $ L^{p}$


Author: Antonios D. Melas
Journal: Proc. Amer. Math. Soc. 147 (2019), 3367-3375
MSC (2010): Primary 42B25
DOI: https://doi.org/10.1090/proc12740
Published electronically: April 18, 2019
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Abstract: We prove a stability estimate for the functions that are almost extremals for the Bellman function related to the $ L^{p}$ norm of the dyadic maximal operator in the case $ p\geq 2$. This estimate gives that such almost extremals are also almost ``eigenfunctions'' for the dyadic maximal operator, in the sense that the $ L^{p}$ distance between the maximal operator applied to the function and a certain multiple of the function is small.


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Antonios D. Melas
Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
Email: amelas@math.uoa.gr

DOI: https://doi.org/10.1090/proc12740
Keywords: Bellman, dyadic maximal, extremals
Received by editor(s): April 9, 2014
Published electronically: April 18, 2019
Additional Notes: This research has been co-financed by the European Union and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF). ARISTEIA I, MAXBELLMAN 2760, research number 70/3/11913
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2019 American Mathematical Society