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Семинар лаборатории теории функций "Современные проблемы комплексного анализа"
30 мая 2019 г. 12:15, г. Ташкент, Национальный университет Узбекистана, Математический факультет, аудитория А-304 (ул. Университетская, 4)


Individual ergodic theorems for infinite measure

S. N. Litvinova, V. I. Chilinb

a Pennsylvania State University, Department of Mathematics
b National University of Uzbekistan named after M. Ulugbek, Tashkent

Аннотация: Given a $\sigma$-finite measure space $(\Omega,\mu)$, it is shown that any Dunford-Schwartz operator $T:\, \cal L^1(\Omega)\to\mathcal L^1(\Omega)$ can be uniquely extended to the space $\mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$. This allows to find the largest subspace $\mathcal R_\mu$ of $\mathcal L^1(\Omega)+\mathcal L^\infty(\Omega)$ such that the ergodic averages $\frac{1}{n}\sum\limits_{k=0}^{n-1}T^k(f)$ converge almost uniformly (in Egorov's sense) for every $f\in\cal R_\mu$ and every Dunford-Schwartz operator $T$. Utilizing this result, almost uniform convergence of the averages $\frac{1}{n}\sum\limits_{k=0}^{n-1}\beta_kT^k(f)$ for every $f\in\mathcal R_\mu$, a Dunford-Schwartz operator $T$, and any bounded Besicovitch sequence $\{\beta_k\}$ is established. Further, given a measure preserving transformation $\tau:\Omega\to\Omega$, Assani's extension of Bourgain's Return Times theorem to $\sigma$-finite measure is employed to show that for each $f\in\mathcal R_\mu$ there exists a set $\Omega_f\subset\Omega$ such that $\mu(\Omega\setminus\Omega_f)=0$ and the averages $\frac1n\sum\limits_{k=0}^{n-1}\beta_kf(\tau^k\omega)$ converge for all $\omega\in\Omega_f$ and any bounded Besicovitch sequence $\{\beta_k\}$. Applications to fully symmetric subspaces $E\subset\mathcal R_\mu$ are given.


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