Square Functions for Ritt Operators in $L^1$

$T$ is a Ritt operator in $L^p$ if ${\sup }_{n}n\mathrm{\parallel }{T}^n-T^{n+1}\mathrm{\parallel }<\mathrm{\infty }$. From \cite{LeMX-Vq}, if $T$ is a positive contraction and a Ritt operator in $L^p$, $1<p<\mathrm{\infty }$, the square function ${({\sum }_n{n}^{2m+1}\mathrm{\mid }{T}^n(I-T{)}^{m+1}f{\mathrm{\mid }}^2)}^{1\mathrm{/}2}$ is bounded. We show that if $T$ is a Ritt operator in $L^1$, $$Q_{\alpha ,s,m}f={(\sum _n{n}^{\alpha }\mathrm{\mid }{T}^n(I-T{)}^{m}f{\mathrm{\mid }}^s)}^{1\mathrm{/}s}$$ is bounded $L^1$ when $\alpha +1<sm$, and examine related questions on variational and oscillation norms.