Eigenfunctions of the Hardy-Littlewood maximal operator

Resumen

In this paper the authors prove that peak-shaped eigenfunctions of the one-dimensional uncentered Hardy-Littlewood maximal operator are symmetric and homogeneous, from which it follows that the norms of the maximal operator on Lp spaces are not attained. In particular they prove the following: Let f(x) be a locally integrable function on a finite or infinite interval a<x<b, with a single peak at a point c. Assume that f(x) is an eigenfunction of the uncentered maximal operator with eigenvalue λ, and let 1<λ, p<∞ be related by the equation(p−1)λp−pλp−1−1=0.Then (a) If c=a,b, then f(x)=d|x−c|(1−λ)/λ for some d>0. (b) If a<c<b and |x−c|<min{c−a,b−c}, then f(x)=d|x−c|−1/p. Moreover, the authors prove the following theorem: (1) The norm of the maximal operator on Lp, 1<p<∞, is the positive solution to the equation(p−1)λp−pλp−1−1=0. (2) This norm is not attained, that is, for every nonzero function f,{∫∞−∞|Mf(x)|pdx}1/p<λ{∫∞−∞|f(x)|pdx}1/p.