On dimension-free variational inequalities for averaging operators in $\mathbb R^d$

TL;DR

This paper investigates dimension-independent $L^p$ bounds for the variation of Hardy--Littlewood averaging operators over symmetric convex bodies in high-dimensional Euclidean spaces.

Contribution

It establishes new dimension-free variational inequalities for averaging operators, extending previous results to a broader class of convex bodies.

Findings

Proves dimension-free bounds for $r$-variations in $L^p$ spaces.

Extends variational inequalities to symmetric convex bodies.

Provides tools for analyzing high-dimensional averaging operators.

Abstract

We study dimension-free $L^p$ inequalities for $r$-variations of the Hardy--Littlewood averaging operators defined over symmetric convex bodies in $\mathbb R^d$.