What does this research mean for the field?

Essentially optimal L^p(R)-estimates hold for variational variants of maximal Fourier multiplier operators, extending the pointwise convergence of ergodic averages of integer parts of real-variable polynomials to a broader class of functions. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Establish essentially optimal Lp-estimates for variational variants of maximal Fourier multiplier operators and extend pointwise ergodic convergence to integer parts of real-variable polynomials.

September 2, 2026Journal of Fourier Analysis and ApplicationsOpen Access

Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

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BKBen KrauseUniversity of Bristol

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Implication

Mathematical analysis demonstrates optimal variational Fourier bounds in measure-preserving systems, indicating broadened pointwise convergence for polynomial ergodic averages.

Key Points

Establish essentially optimal Lp-estimates for variational variants of maximal Fourier multiplier operators and extend pointwise ergodic convergence to integer parts of real-variable polynomials.
Analyzed variational variants of Bourgain's maximal Fourier multiplier operators using multi-frequency oscillation methods.
Evaluated discrete ergodic averages along integer parts of real-variable polynomials, floor(P(n)), acting on sigma-finite measure spaces with measure-preserving transformations.
Proved essentially optimal Lp(R)-estimates for variational extensions of maximal Fourier multiplier operators.
Demonstrated almost everywhere convergence of ergodic averages for linear polynomials when 1 < p ≤ ∞.
Established almost everywhere convergence of ergodic averages for nonlinear real polynomials when 4/3 < p ≤ ∞.

Cite This Study

Ben Krause (2026) studied this question.

synapsesocial.com/papers/6a97e20ec562ede874ec60f7 https://doi.org/10.1007/s00041-026-10297-5

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Implication

Mathematical analysis demonstrates optimal variational Fourier bounds in measure-preserving systems, indicating broadened pointwise convergence for polynomial ergodic averages.

Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

Discussion

Member takes

Implication

Key Points

Cite This Study

Multi-Frequency Oscillation Estimates Arising in Pointwise Ergodic Theory

Discussion

Member takes

Implication

Key Points

Cite This Study

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