$T$ 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.
Categories: math.SP