Square Functions for Ritt Operators in $L^1$

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.