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Convergence of weighted polynomial multiple ergodic averages. (English) Zbl 1175.37003
Let $(Y,\nu)$ be a probability space with measure $\nu$, and let $\{b_n\}$ be a bounded sequence of real numbers $b_n(n\in Z)$. The sequence $\{b_n\}$ is called to be universally good for the convergence in the mean of polynomial multiple ergodic averages (in short, for c.m.p.m.e.a.), if for any system $(X,\mu,T)$, for any $r\ge 1$, all polynomials with integer coefficients $p_1,p_2,\dots,p_r$ and all $f_1, \dots,f_r\in L^\infty(\mu)$ the averages $(1/N)\sum^{N-1}_{n=0}b_n T^{p_1(n)}f_1\dots T^{p_r(n)}f_r$ converges in $L^2(\mu)$ as $N\to \infty$. The author proves the following theorem: Let $(Y,\nu,S)$ be an ergodic system and $\varphi\in L^\infty(\nu)$. Then there exists $Y_0 \subset Y$ with $\nu(Y_0)=1$ such that for every $y_0\in Y_0$ the sequence $b_n=\varphi(S^ny_0)$ $(n\in Z)$ is universally good for c.m.p.m.e.a. . By making use of the generalized Wiener-Wintner theorem proved by Host and Kra, the author proves the main theorem by proving the convergence criterion for weighted averages.

37A05Measure-preserving transformations
37A30Ergodic theorems, spectral theory, Markov operators
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