Extremal sequences for the Bellman function of the dyadic maximal operator

Abstract

We give a characterization of the extremal sequences for the Bellman function of the dyadic maximal operator. In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue.

A sharp integral inequality for the dyadic maximal operator and a related stability result

“…By using now the results of [10] we conclude that if we fix the L 1 and L p norms of φ n , n ∈ N, the sequence (φ n ) n gives equality in the limit in (1.8) for any q ∈ (1, p], if and only if it behaves as an extremal sequence for the respective Bellman function (1.6). At last we mention that the evaluation of (1.6) has been given by an alternative method in [13] while certain Bellman functions corresponding to several problems in harmonic analysis, have been studied in [6], [7], [14], [15], [16] and [17].…”

“…T (f, A). By using the results of [10] we get that all such sequences behave like L q -approximate eigenfunctions for the eigenvalue ω q ( f q A ) which equals β + 1. That is the following holds…”

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

“…In this direction it has been proved in [9] (for …xed F; f; p) that if a sequence ( n ) of nonnegative functions is extremal for (1.8) in the sense that R…”

A multiparameter integral inequality for the dyadic maximal operator and applications

“…We also need to mention that the extremizers for the standard Bellman function B T p (f, F, f, 1) has been studied in [16], and in [18] for the case 0 < p < 1. We note also that further study of the dyadic maximal operator can be seen in [19,20] where symmetrization principles for this operator are presented, while other approaches for the determination of certain Bellman functions are given in [26,27,31,32,33] .…”