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Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain
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abstract
Let $L^2(X,\Sigma,\mu,\tau)$ be a measure-preserving system, with $\tau$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} \tau^{P(n)} f \] converge pointwise for $f \in L^2(X)$. We do so by proving that, for $r>2$, the $r$-variation, $\mathcal{V}^r(M_N(f))$, extends to a bounded operator on $L^2$. We also prove that our result is sharp, in that $\mathcal{V}^2(M_N(f))$ is an unbounded operator on $L^2$.
Forward citations
Cited by 2 Pith papers
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Discrete analogues in harmonic analysis: Multi-parameter Radon averages
Every multi-parameter polynomial ergodic average on a measure space with commuting transformations converges almost everywhere; the proof gives sharp ℓ^p oscillation and maximal inequalities for discrete Radon averages.
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A Variation Norm Carleson Theorem Along the Primes
The variational prime Carleson operator is bounded on ℓ^p for an r-dependent range c(r)<p<C(r) that approaches the full (1,∞) as r o∞, and the maximal version is bounded for all 1<p<∞.
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