A rigidity phenomenon for the Hardy-Littlewood maximal function

Stefan Steinerberger. "A rigidity phenomenon for the Hardy-Littlewood maximal function." Studia Mathematica 229.3 (2015): 263-278. <http://eudml.org/doc/285673>.

@article{StefanSteinerberger2015,
abstract = {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 $f ∈ C^\{α\}(ℝ,ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $(A_\{x\}f)(r) = 1/2r ∫_\{x-r\}^\{x+r\} f(z)dz$ 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.},
author = {Stefan Steinerberger},
journal = {Studia Mathematica},
keywords = {averaging operator; Hardy-Littlewood maximal function; transcendental number theory; Lindemann-Weierstrass theorem; delay differential equation},
language = {eng},
number = {3},
pages = {263-278},
title = {A rigidity phenomenon for the Hardy-Littlewood maximal function},
url = {http://eudml.org/doc/285673},
volume = {229},
year = {2015},
}

TY - JOUR
AU - Stefan Steinerberger
TI - A rigidity phenomenon for the Hardy-Littlewood maximal function
JO - Studia Mathematica
PY - 2015
VL - 229
IS - 3
SP - 263
EP - 278
AB - 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 $f ∈ C^{α}(ℝ,ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $(A_{x}f)(r) = 1/2r ∫_{x-r}^{x+r} f(z)dz$ 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.
LA - eng
KW - averaging operator; Hardy-Littlewood maximal function; transcendental number theory; Lindemann-Weierstrass theorem; delay differential equation
UR - http://eudml.org/doc/285673
ER -