Square Functions for Ritt Operators in $L^1$

Abstract

$T$ is a Ritt operator in $L^p$ if ${\sup }_{n}n\Vert T^n-T^{n+1}\Vert <\infty$. 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 $L^1$, \[Q_{α,s,m}f=\left( \sum_n n^α |T^n(I-T)^mf|^s \right)^{1/s}\] is bounded $L^1$ when $\hat{I}\pm +1<sm$, and examine related questions on variational and oscillation norms.