The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as those nonconstant functions for which the computation of the maximal function ℳ is as simple as possible. The proof uses the Lindemann-Weierstrass theorem from transcendental number theory.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm8368-12-2015,
author = {Stefan Steinerberger},
title = {A rigidity phenomenon for the Hardy-Littlewood maximal function},
journal = {Studia Mathematica},
volume = {231},
year = {2015},
pages = {263-278},
zbl = {1337.42022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8368-12-2015}
}
Stefan Steinerberger. A rigidity phenomenon for the Hardy-Littlewood maximal function. Studia Mathematica, Tome 231 (2015) pp. 263-278. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm8368-12-2015/