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

TL;DR

This paper explores bounds and counterexamples for the second derivative of the uncentered Hardy-Littlewood maximal function in one dimension, focusing on functions with specific symmetry and monotonicity properties.

Contribution

It establishes positive bounds for certain symmetric decreasing functions and provides counterexamples for non-symmetric decreasing functions regarding the second derivative's $L^1$ norm.

Findings

Bounded the second derivative for symmetric decreasing functions.

Constructed counterexamples for non-symmetric decreasing functions.

Clarified conditions under which the second derivative norm can be controlled.

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.