Square functions and variational estimates for Ritt operators on $L^1$

Abstract A power-bounded operator T satisfying $\sup _n n\lVert T^n-T^{n+1}\rVert <\infty $ is a Ritt operator. For such operators, we study the generalized square function $Q_{\alpha ,s,r}^Tf=(\sum _n n^{\alpha } |T^n(I-T)^rf|^s)^{1/s}$ . It is known that when T is a positive contraction and a Ritt operator on $L^p$ , $1<p<\infty $ , then for any integer $r\ge 1$ , the square function $Q_{2r-1,2,r}^Tf$ defines a bounded operator [17] on $L^p$ . In this work, we extend the theory to the endpoint case $p=1$ . We show that if T is a Ritt operator on $L^1$ , then the generalized square function $Q_{\alpha ,s,r}^Tf $ is bounded on $L^1$ whenever $\alpha +1<sr$ . In the particular 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 measure-preserving transformation, we provide sufficient conditions on $\mu $ under which $Q_{2r-1,2,r}^{T_{\mu }}f$ is of weak type $(1,1)$ , for $r>0$ . 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 en