A rigidity phenomenon for the Hardy–Littlewood maximal function
Volume 229 / 2015
Abstract
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.
