A smoother ergodic average (Q1593030)

Let \(\{T_t\}_{t\in\mathbb{R}}\) be a measure preserving flow on a probability space \((X,\beta, m)\). Let \((\varepsilon_k)\) be a sequence in \((0,1)\), and let \(\phi\) be a positive integrable function on \(\mathbb{R}\) satisfying (i) \(\int\phi(x) dx= 1\), and (ii) the function \(\Phi(x)= \sup_{|y|\geq|x|}\phi(y)\) is integrable. In this paper, the author investigates the convergence of the sequence \[ P_n f(x)= {1\over n} \sum^n_{k=1} {1\over\varepsilon_k} \int f(T_{t+k}x) \phi\Biggl({1\over \varepsilon_k}\Biggr) dt \] for \(f\in L^p\), \(1\leq p\leq\infty\). She proves that if \(\lim_{k\to\infty} \varepsilon_k= \varepsilon\geq 0\), then \((P_nf)\) converges in \(L^p\). She also studies the convergence of these averages in case the sequence \((\varepsilon_k)\) is not convergent but satisfies the variation condition \[ \lim_{n\to\infty} {1\over n} \sum^n_{k=0}|\varepsilon_{k-1}- \varepsilon_k|/(\varepsilon_{k-1}\vee \varepsilon_k)= 0. \] She proves that \(T_1\) is ergodic if and only if for any sequence \((\varepsilon_k)\) satisfying the above variation conditions one has, (a) if \(p> 1\), then \(\lim_{n\to\infty} P_nf\) exists a.e. for any \(f\in L^p\), and (b) if \(p=1\) and \(\varepsilon_k\geq \varepsilon> 0\) for all \(k\), then \(\lim_{n\to\infty} P_nf\) exists a.e. for any \(f\in L^1\). In the last section of this paper, the convergence of these averages are studied in case the sequence \((\varepsilon_k)\) is given by a stationary sequence of random variables.