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A rigidity phenomenon for the Hardy-Littlewood maximal function

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Steinerberger S.

Studia Mathematica

2015

tom 229

nr 3

263-278

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.

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1 Studia Mathematica

1 Steinerberger S.

1 2015

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A rigidity phenomenon for the Hardy-Littlewood maximal function