Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain

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, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} τ^{P(n)} f \] 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 $L^2$.

23 pages

References in corpus (1)

• Some Optimizations for (Maximal) Multipliers in $L^p$

Cited by in corpus (12)

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• Jump inequalities for translation-invariant operators of Radon type on ${\mathbb{Z}}^d$
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• L^p(Z^d)-estimates for discrete operators of Radon type: Variational estimates
• Discrete Analogoues in Harmonic Analysis: Maximally Monomially Modulated Singular Integrals Related to Carleson's Theorem
• Some jump and variational inequalities for the Calderón commutators and related operators
• Variational inequalities for the commutators of rough operators with BMO functions
• A Discrete Quadratic Carleson Theorem on ${\ell }^2$ with a Restricted Supremum
• Discrete bilinear Radon transforms along arithmetic functions with many common values
• Pointwise ergodic theorems for some thin subsets of primes
• A variable coefficient multi-frequency lemma