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Variation-norm and fluctuation estimates for ergodic bilinear averages

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Yen Do, Richard Oberlin, Eyvindur A. Palsson

Classical Analysis and ODEs · Submitted 27 Apr 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates.

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arXiv ID1504.07134 (version 2)
Pith IDVQZ3DCMY, Pith's permanent record of this paper and its review
CitationEyvindur A. Palsson, Richard Oberlin, Yen Do. "Variation-norm and fluctuation estimates for ergodic bilinear averages." arXiv:1504.07134 (2015). https://pith.science/paper/VQZ3DCMY
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arXiv listing abstract v2. Results and the main claim are underlined; .

For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variation-norms of truncated bilinear operators on R, and the main ingredient of the proof of these bounds is a variation-norm extension of maximal Bessel inequalities of Lacey and Demeter--Tao--Thiele.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. result 0.77 0.95 0.77 asserted 1.00
s1 It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. result 0.59 0.84 0.25 hedged possibility 0.80
s2 These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). interpretation 0.61 0.23 0.13 asserted 1.00
s3 Via transference, the proof reduces to establishing new bilinear Lp bounds for variation-norms of truncated bilinear operators on R, and the main ingredient of the proof of these bounds is a variation-norm extension of maximal Bessel inequalities of Lacey and Demeter--Tao--Thiele. method 0.99 0.85 0.07 asserted 1.00

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Original abstract

For any dynamical system, we show that higher variation-norms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variation-norms of truncated bilinear operators on R, and the main ingredient of the proof of these bounds is a variation-norm extension of maximal Bessel inequalities of Lacey and Demeter--Tao--Thiele.

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