Variation bounds for spherical averages over restricted dilates

TL;DR

This paper investigates bounds on the variation semi-norms of spherical averaging operators over restricted dilates, providing new insights into their behavior in certain function spaces.

Contribution

It introduces novel variation bounds for spherical averages over restricted dilates, extending previous understanding of these operators in harmonic analysis.

Findings

Established $L^p$ to $L^q$ variation bounds for spherical averages

Extended analysis to restricted dilates within subset $E$ of [1,2]

Provided new estimates for variation semi-norms in harmonic analysis

Abstract

We study $L^p\rightarrow L^q(V^r_E)$ variation semi-norm estimates for the spherical averaging operator, where $E\subset [1,2]$.