Polynomial averages and pointwise ergodic theorems on nilpotent groups

Abstract

We establish pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two of measure-preserving transformations on σ-finite measure spaces. We also establish corresponding maximal inequalities on L p for 1 < p ≤ ∞ and ρ-variational inequalities on L 2 for 2 < ρ < ∞. This gives an affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the linear case for all polynomial ergodic averages in discrete nilpotent groups of step two.Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative, nilpotent setting. In particular, we develop what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups. Contents 1. Introduction 1 2. Notation and preliminaries 3. Ergodic theorems: Proof of Theorem 1.2 4. Maximal and variational estimates on G 0 : ℓ 2 theory 5. Minor arcs contributions:

Discrete Double Hilbert Transforms Along Polynomial Surfaces

“…Moreover, in 2007, Ionescu, Magyar, Stein and Wainger [12] establshed the ℓ p (Z d ) theorems for the non-translation invariant singular integral and maximal average and the corresponding ergodic theorem when deg(P ) ≤ 2. Recently, Ionescu, Magyar, Mirek and Szarek [14] proved partial cases of the Furstenberg-Bergelson-Leibman conjecture regarding the pointwise ergodic theorems on nilpotent groups.…”