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We prove a stability estimate for the functions that are almost extremals for the Bellman function related to the Lp norm of the dyadic maximal operator in the case p 2. This estimate gives that such almost extremals are also almost "eigenfunctions" for the dyadic maximal operator, in the sense that the Lpdistance between the maximal operator applied to the function and a certain multiple of the function is small. Acknowledgement 1. This research has been co-nanced 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. 1. Introduction The dyadic maximal operator on R is a useful tool in analysis and is dened by (1.1) M d (x) = sup 1 jQj Z Q j (u)j du : x 2 Q, Q R is a dyadic cube for every 2 Lloc(R) where the dyadic cubes are the cubes formed by the grids 2 Z for N = 0; 1; 2; :::. As it is well known it satises the following L inequality (1.2) kMd kp p p 1 k kp for every p > 1 and every 2 L(R) which has been proved best possible ([2] for the general martingales and [17] for dyadic ones). In studying dyadic maximal operators as well as more general variants it would be convenient to work with functions supported in the unit cube [0; 1] and replace Md by (1.3) M 0 d (x) = sup 1 jQj Z Q j (u)j du : x 2 Q, Q [0; 1] is a dyadic cube and hence work completely on the measure space [0; 1] and actually we can work on a general nonatomic probability space with a martingale structure similar to the dyadic one. Date : Dec. 5, 2013. 1991 Mathematics Subject Classication. [2010] 42B25.

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