{"paper":{"title":"Mean ergodic theorems in $L^r(\\mu)$ and $H^r(\\mathbb T)$, $0<r<1$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.DS","authors_text":"el Houcein el Abdalaoui, Michael Lin","submitted_at":"2023-12-31T19:05:09Z","abstract_excerpt":"Let $T$ be the Koopman operator of a measure preserving transformation $\\theta$ of a probability space $(X,\\Sigma,\\mu)$. We study the convergence properties of the averages $M_nf:=\\frac1n\\sum_{k=0}^{n-1}T^kf$ when $f \\in L^r(\\mu)$, $0<r<1$. We prove that if $\\int |M_nf|^r d\\mu \\to 0$, then $f \\in \\overline{(I-T)L^r}$, and show that the converse fails whenever $\\theta$ is ergodic aperiodic. When $\\theta$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r \\in (I-T)L^r$ for which $M_nf_r$ does not converge a.e. (although $\\int |M_nf|^r d\\mu \\to 0$). We further establish t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2401.00567","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2401.00567/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}