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

Abstract

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 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. Acknowledgement 1.

Stability in Burkholder's differentially subordinate martingales inequalities and applications to Fourier multipliers

“…A careful inspection of the paper [36] reveals that for any ε > 0 there is f ∈ L p for which the pointwise identity Mf = p p−1 − ε f holds true and therefore this family of functions, corresponding to different ε, can be regarded as an "approximate eigenfunction" of M associated with the eigenvalue p/(p − 1). One of the main results of [37] makes this observation more precise. It is proved that if 2 < p < ∞ is a fixed exponent, ε > 0 is a small number and f is any nonnegative function satisfying…”

“…Our results are motivated from the recent paper [37] by Melas concerning the structure of almost extremal functions associated with the L p -estimate for the dyadic maximal operator M on [0, 1] d , a version of Doob's maximal inequality. It is well-known that M is a bounded operator on L p ([0, 1] d ), 1 < p < ∞, and its norm equals p/(p − 1).…”