Square Functions and Variational Estimates for Ritt Operators on $L^1$

Let $T$ be a bounded operator. We say $T$ is a Ritt operator if $\sup_n n\lVert T^n-T^{n+1}\rVert<\infty$. It is know that when $T$ is a positive contraction and a Ritt operator in $L^p$, $1<p<\infty$, then for any integer $m\ge 1$, the square function $$\Big( \sum_n n^{2m-1} |T^n(I-T)^{m}f|^2 \Big)^{1/2}$$ defines a bounded operator \cite{LeMX-Vq} in $L^p$. In this work, we extend the theory to the endpoint case $p=1$, showing that if $T$ is a Ritt operator on $L^1$, then the generalized square function $$Q_{\alpha,s,m}f=\Big( \sum_n n^{\alpha} |T^n(I-T)^mf|^s \Big)^{1/s}$$ is bounded on $L^1$ for $\alpha+1<sm$. In the specific setting where $T$ is a convolution operator of the form $T_{\mu}=\sum_k \mu(k) U^kf$, with $\mu$ a probability measure on $\mathbb Z$ and $U$ the composition operator induced by an invertible, ergodic measure preserving transformation, we provide sufficient conditions on $\mu$ under which the square function $Q_{2m-1,2,m}$ is of weak type (1,1), for all integers $m\ge 1$. We also establish bounds for variational and oscillation norms, $\lVert n^{\beta} T^n(1-T)^r\rVert_{v(s)}$ and $\lVert n^{\beta} T^n(1-T)^r\rVert_{o(s)}$, for Ritt operators, highlighting endpoint behavior.