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

Abstract

We prove a stability estimate for the functions that are almost extremals for the Bellman function related to the L p L^{p} norm of the dyadic maximal operator in the case p ≥ 2 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 L^{p} distance between the maximal operator applied to the function and a certain multiple of the function is small.