Discrete Harmonic Analisys and its applications to Ergodic Theory

DATES: 22-26 November 2021 (4 sessions)
TIME: 9:30 - 11:30 (a total of 10 hours)*
LOCATION: BCAM and Online

*There has been a time change, the course will start and end 30 minutes later

ABSTRACT:
The aim of this minicourse is to show the applications of the discrete harmonic analysis to (pointwise) ergodic theory. Let k, d ∈ Z+ and let (X, μ) be a σ-finite measure space. Given any family T1, . . ., Td: X → X of commuting, invertible measure-preserving transformations, a measurable function f, polynomials P1, . . ., Pd: Zk → Zd, and a natural number N ≥ 1, we can define a polynomial ergodic average by the formula
(0.1)

AP1,...,Pd
N;X,T1,...,Td (f)(x) := 1/|[−N,N]k ∩ Zk| X n∈[−N,N]k∩Z f(TP1(n) 1  TPd(n) d x), x ∈ X.

The classical question in ergodic theory is whether the means in (0.1) converge pointwise or/and in Lp norm, 1 ≤ p < ∞. The answer to this question is affirmative. Convergence in norm was shown in [5] in a more general (noncommutative) situation. During the course we focus mainly on pointwise (almost everywhere) convergence, which is more involved but also solved question. We first give a classical proof of the special case d = k = 1 and P1(n) = n, which is the classical Birkhoff's pointwise ergodic theorem, see [1]. This question was solved for d = k = 1 and a general polynomial P1 by Bourgain in a series of papers in 80´s and recently by Mirek, Stein and Trojan in [3] in the full generality.
During the lectures we show that using the Calderón principle from [2] we can reduce our considerations to proving qualitatively bounds corresponding to the special case where X = Zd, μ is the counting measure and Tjx = x − ej , 1 ≤ j ≤ d, are shift operators. We will introduce the notion of jump and variational estimates, which are very useful in problems related to pointwise convergence, and we present the ideas which allow us to obtain the required bounds in this special dynamical system on Zd.
If time permits we want to consider also discrete spherical maximal function defined by defined by
MSf(x) = sup λ∈N 1 N(λ) X y∈N(λ) f(x − y),
where N(λ) = {x ∈ Zd : x21+ . . . + x2d = λ}. This part would be based on the paper [4], where characterization of lp(Zd)-boundedness of these operators was shown.

PREREQUISITES:
None, but basic knowledge concerning classical real analysis would be desirable.

REFERENCES:
[1] G. Birkhoff, Proof of the ergodic theorem, Proc. Natl. Acad. Sci. USA 17 (1931), 656-660.
[2] A. Calderón, Ergodic theory and translation invariant operators, Proc. Natl. Acad. Sci. USA 59 (1968), 349-353.
[3] M. Mirek, E.M. Stein, B. Trojan, lp(Zd)-estimates for discrete operators of Radon type: Variational estimates, Invent. Math. 209 (2017), 665-748.
[4] A. Magyar, E.M. Stein, S. Wainger, Discrete analogues in harmonic analysis: spherical averages, Ann. Math. 155 (2002), 189-208.
[5] M. Walsh, Norm convergence of nilpotent ergodic averages, Ann. Math. 175 (2012), 1667-1688.

*Registration is free, but mandatory before November 17th. To sign-up go to https://forms.gle/f4AZb8uqXaQcMyyL7 and fill the registration form.

• BCAM-Course_20211122-26