A Rigidity Phenomenon for the Hardy-Littlewood Maximal Function

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 $$(A_{x}f)(r)=\frac{1}{2r}{\int }_{x-r}^{x+r}f(z)dz$$ has a critical point in $r=\gamma$ for every $x\in \mathbb{R}$, then $$f(x) = a+b\sin{(cx + d)} \qquad \mbox{for some}~a,b,c,d \in \mathbb{R}.$$ 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.

to appear in Studia Mathematica