Square Functions for Ritt Operators in $L^1$

TL;DR

This paper investigates square functions associated with Ritt operators in $L^1$, establishing boundedness conditions and exploring related variational and oscillation norm questions.

Contribution

It extends the theory of Ritt operators to the $L^1$ setting, providing new boundedness results for associated square functions and related norms.

Findings

Square functions are bounded in $L^1$ under certain parameter conditions.

Conditions for boundedness depend on the relation $ ext{alpha}+1<sm$.

Explores variational and oscillation norm properties for Ritt operators.

Abstract

$T$ is a Ritt operator in $L^p$ if $\sup_n n\|T^n-T^{n+1}\|<\infty$. From \cite{LeMX-Vq}, if $T$ is a positive contraction and a Ritt operator in $L^p$, $1<p<\infty$, the square function $\left( \sum_n n^{2m+1} |T^n(I-T)^{m+1}f|^2 \right)^{1/2}$ is bounded. We show that if $T$ is a Ritt operator in $L^1$, \[Q_{\alpha,s,m}f=\left( \sum_n n^{\alpha} |T^n(I-T)^mf|^s \right)^{1/s}\] is bounded $L^1$ when $\alpha+1<sm$, and examine related questions on variational and oscillation norms.