A rigidity phenomenon for the Hardy-Littlewood maximal function (Q2787138)

The main purpose of this paper is to show that if a periodic function \(f\) is sufficiently smooth, then there is a positive number \(z\) such that the averaging function NEWLINE\[NEWLINE A_{x}(f)(r)=\frac{1}{2r}\int_{x-r}^{x+r}f(y)dy NEWLINE\]NEWLINE has a critical point at \(r=z\) for every \(x\) if and only if \(f(x)=a+b\sin (cx+d)\). Somewhat unexpectedly, the proof involves transcendental number theory. There is a delay differential equation naturally associated with the averaging operator and it easily follows from the main result that the only periodic solution to this delay differential equation is the function \(f\). Another application is to show that if the maximal function \(Mf(x)\) satisfies the simple equation NEWLINE\[NEWLINE Mf(x)=\max \left( |f(x)|,\frac{1}{2z}\int_{x-z}^{x+z}|f|\right) \text{ for all }x NEWLINE\]NEWLINE and similarly for \(M(-f)\), then again \(f(x)=a+b\sin (cx+d)\).