Adjacent dyadic systems and the $L^p$-boundedness of shift operators in metric spaces revisited

TL;DR

This paper provides an alternative proof for the $L^p$-boundedness of shift operators on functions in metric spaces with UMD spaces, utilizing recent adjacent dyadic constructions.

Contribution

It introduces a new proof method leveraging recent dyadic constructions to establish $L^p$-boundedness results for shift operators in metric spaces.

Findings

Established $L^p$-boundedness of shift operators using dyadic techniques

Extended previous results to more general metric spaces

Provided a new perspective on operator boundedness proofs

Abstract

With the help of recent adjacent dyadic constructions by Hyt\"onen and the author, we give an alternative proof of results of Lechner, M\"uller and Passenbrunner about the $L^p$-boundedness of shift operators acting on functions $f \in L^p(X;E)$ where $1 < p < \infty$, $X$ is a metric space and $E$ is a UMD space.