Year: 2026
Type: preprint
Abstract: We consider the central Hardy--Littlewood maximal operator in $\mathbb{R}^n$. For every integer $n\geq1$, we construct two bounded, nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ that differ on a set of positive measure and satisfy $M_cf(x)=M_cg(x)$ at every point $x\in\mathbb{R}^n$. Thus the central maximal operator is not injective on the nonnegative cone of $L^1(\mathbb{R}^n)$. The functions can have compact support; when $n\geq2$, they can also be chosen radial, either with compact support or strictly positive throughout the space. The construction makes a small redistribution of mass invisible to the maximal operator by excluding averaging spheres that cross its support. We also give an explicit formula, in terms of the incomplete beta function, for the $n$-dimensional volume of the intersection of two balls with arbitrary radii and centers, and an elementary formula for the central maximal function of a ball in $\mathbb{R}^3$.
Source: arXiv (Cornell University)
Author: Solyanik, Alexey
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214176456