A rigidity phenomenon for the Hardy-Littlewood maximal function

• Autorzy: Stefan Steinerberger
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2015
• Tom: 229
• Numer: 3
• Strony: 263-278

Daty

wydano

2015

Twórcy

autor

Stefan Steinerberger

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

Identyfikatory

DOI

10.4064/sm8368-12-2015