Sobolev bounds and counterexamples for the second derivative of the maximal function in one dimension

TL;DR

The paper investigates whether the $L^1$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function is controlled by the $L^1$-norm of the original function. It establishes a positive result for a Sobolev-class of functions that are even and nonincreasing away from the origin, and provides a counterexample outside this class, showing the bound cannot hold in full generality. The authors develop a robust approximation framework to transfer smooth bounds to general weakly differentiable inputs, using a detailed analysis of the maximizing interval endpoints and derivative formulas inspired by Luiro’s representation. These results clarify the limitations and possibilities of second-derivative (regularity) estimates for maximal functions in one dimension, with implications for endpoint variation theory and the role of symmetry in extremizers. The work highlights a nuanced landscape: while radial/even symmetry yields favorable bounds for the uncentered operator, general functions—even in 1D—can exhibit unbounded second-derivative variation, underscoring the delicate balance between regularity and maximal-operator geometry.

Abstract

We investigate the question whether the $L^1(\mathbb R)$-norm of the second derivative of the uncentered Hardy-Littlewood maximal function can be bounded by a constant times the $L^1(\mathbb R)$-norm of the function itself. We give a positive answer for a class of functions that contains Sobolev functions on the real line which are decreasing away from the origin and even, and we provide a counterexample which is also decreasing away from the origin but not even.

Theorems & Definitions (49)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• proof : Proof of \ref{['theorem_radial']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof