Discrete maximal functions in higher dimensions and applications to ergodic theory

Abstract

Abstract. We establish a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates Vr on L p spaces for all 1 < p < ∞ and r > max{p, p/(p − 1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

Maximal function inequalities and a theorem of Birch

“…Similarly, the methods of Magyar, Stein, and Wainger apply to give the result for indefinite quadratic forms of rank at least 5 in the range p > n/(n − 2). Also note that the results of [8] cover special cases of our main result in the full range p > 1.…”

Dimension-Free Estimates for Discrete Hardy-Littlewood Averaging Operators Over the Cubes in ℤd

“…Inequality (1.21) replaces the fractional integration argument from [8] (as it is not clear if this argument is available in the discrete setting) and allows us to obtain (1.12) for p ∈ (3/2, 2]. A variant of this inequality was proven by Lewko-Lewko [12,Lemma 13] in the context of variational Rademacher-Menshov type results for orthonormal systems and it was also obtained independently by the second author and Trojan [15,Lemma 1] in the context of variational estimates for discrete Radon transforms, see also [14]. Inequality (1.21) reduces estimates for a supremum or an r-variation restricted to a dyadic block to the situation of certain square functions, where the division intervals over which differences are taken (in these square functions) are all of the same size.…”

A bootstrapping approach to jump inequalities and their applications

“…Lemma 2.5 originates in the paper of Lewko and Lewko [LL12], where it was observed that the 2-variation norm of a sequence of length N can be controlled by the sum of log N square functions and this observation was used to obtain a variational version of the Rademacher-Menshov theorem. Inequality (2.6), essentially in this form, was independently proved by the first author and Trojan in [MT16] and used to estimate r-variations for discrete Radon transforms. Lemma 2.5 has been used in several recent articles on r-variations, including [Bou+18a].…”