Square Functions and Variational Estimates for Ritt Operators on $L^1$
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866914495768559616 |
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| author | Hults, Jennifer Reinhold-Larsson, Karin |
| author_facet | Hults, Jennifer Reinhold-Larsson, Karin |
| contents | 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 \[Q_{α,s,m}f=\Big( \sum_n n^α |T^n(I-T)^mf|^s \Big)^{1/s}\] is bounded on $L^1$ for $α+1<sm$.
In the specific setting where $T$ is a convolution operator of the form $T_μ=\sum_k μ(k) U^kf$, with $μ$ a probability measure on $\mathbb Z$ and $U$ the composition operator induced by an invertible, ergodic measure preserving transformation, we provide sufficient conditions on $μ$ 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, $\lVert n^β T^n(1-T)^r\rVert_{v(s)}$ and $\lVert n^β T^n(1-T)^r\rVert_{o(s)}$, for Ritt operators, highlighting endpoint behavior. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07256 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Square Functions and Variational Estimates for Ritt Operators on $L^1$ Hults, Jennifer Reinhold-Larsson, Karin Functional Analysis 47A35, 47Axx, 28Dxx, 37Axx, 58J51, 37A30 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 \[Q_{α,s,m}f=\Big( \sum_n n^α |T^n(I-T)^mf|^s \Big)^{1/s}\] is bounded on $L^1$ for $α+1<sm$. In the specific setting where $T$ is a convolution operator of the form $T_μ=\sum_k μ(k) U^kf$, with $μ$ a probability measure on $\mathbb Z$ and $U$ the composition operator induced by an invertible, ergodic measure preserving transformation, we provide sufficient conditions on $μ$ 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, $\lVert n^β T^n(1-T)^r\rVert_{v(s)}$ and $\lVert n^β T^n(1-T)^r\rVert_{o(s)}$, for Ritt operators, highlighting endpoint behavior. |
| title | Square Functions and Variational Estimates for Ritt Operators on $L^1$ |
| topic | Functional Analysis 47A35, 47Axx, 28Dxx, 37Axx, 58J51, 37A30 |
| url | https://arxiv.org/abs/2507.07256 |