A rigidity phenomenon for the Hardy–Littlewood maximal function

Stefan Steinerberger¹

¹ Department of Mathematics Yale University 10 Hillhouse Avenue New Haven, CT 06511, U.S.A.

Studia Mathematica, Tome 229 (2015) no. 3, pp. 263-278

Résumé

The Hardy–Littlewood maximal function $\mathcal {M}$ and the trigonometric function $\sin x$ are two central objects in harmonic analysis. We prove that $\mathcal {M}$ characterizes $\sin x$ in the following way: Let $f \in C^{\alpha }(\mathbb {R}, \mathbb {R})$ be a periodic function and $\alpha > 1/2$. If there exists a real number $0 \gamma \infty $ such that the averaging operator $$ (A_xf)(r) = \frac {1}{2r}\int _{x-r}^{x+r}{f(z)\,dz}$$ has a critical point at $r = \gamma $ for every $x \in \mathbb {R}$, then $$f(x) = a+b\sin (cx + d) \hskip 1em\hbox {for some } a,b,c,d \in \mathbb {R}.$$ This statement can be used to derive a characterization of trigonometric functions as those nonconstant functions for which the computation of the maximal function $\mathcal {M}$ is as simple as possible. The proof uses the Lindemann–Weierstrass theorem from transcendental number theory.

Détail

DOI : 10.4064/sm8368-12-2015

Keywords: hardy littlewood maximal function mathcal trigonometric function sin central objects harmonic analysis prove mathcal characterizes sin following alpha mathbb mathbb periodic function alpha there exists real number gamma infty averaging operator frac int x r has critical point gamma every mathbb sin hskip hbox d mathbb statement derive characterization trigonometric functions those nonconstant functions which computation maximal function mathcal simple possible proof uses lindemann weierstrass theorem transcendental number theory