Variation estimates for averages along primes and polynomials

Abstract

We prove $q$-variation estimates, $q>2$, on ${\ell }^p$ spaces for averages along primes (with $1<p<\infty$) and polynomials (with $|\frac{1}{p}-\frac{1}{2}|<\frac{1}{2(d+1)}$, where $d$ is the degree of the polynomial). This improves the pointwise ergodic theorems for these averages in the corresponding ranges of $L^p$ spaces.

v4: final, revised following referee's suggestions

References in corpus (3)

• Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain
• Some Optimizations for (Maximal) Multipliers in $L^p$
• Discrete maximal functions in higher dimensions and applications to ergodic theory

Cited by in corpus (14)

• Jump inequalities for translation-invariant operators of Radon type on ${\mathbb{Z}}^d$
• Variational estimates for averages and truncated singular integrals along the prime numbers
• Nilsystems and ergodic averages along primes
• L^p(Z^d)-estimates for discrete operators of Radon type: Variational estimates
• Weighted Lépingle inequality
• A Discrete Quadratic Carleson Theorem on ${\ell }^2$ with a Restricted Supremum
• Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes
• Discrete bilinear Radon transforms along arithmetic functions with many common values
• Quantitative weighted bounds for the $q$-variation of singular integrals with rough kernels
• Pointwise ergodic theorems for some thin subsets of primes
• Bootstrap methods in bounding discrete Radon operators
• A variable coefficient multi-frequency lemma
• Variational Inequalities for Bilinear Averaging Operators over Convex Bodies
• Quantitative ergodic theorems for actions of groups of polynomial growth