Oscillation in ergodic theory

Abstract

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

JOP's counting function and Jones' square function

“…The following result was already obtained for p ≥ 2 and d = 1 in [9], and for d > 1 in [10]. Here is a simple proof for 1 < p < ∞ and d ≥ 1.…”

A variation norm Carleson theorem

“…Simple proofs (based on jump inequalities) have been obtained in [1] and [27] (see also [9], [15] for other expositions). Inequality (40) has been extended to various families of convolution operators ( [1], [14], [2], [15]). Let ψ be a Schwartz function on R with ψ = 1, for each k let ψ k = 2 −k ψ(2 −k ·) and let …”

“…From the energy bound (applied to multitiles) and the bound (14) for |φ P |, the right side is bounded by…”

“…Simple proofs of Lépingle's result based on jump inequalities have been given by Pisier and Xu [27] and by Bourgain [1], and applications to other families of operators in harmonic analysis such as families of averages and singular integrals have been considered in [1], and the subsequent papers [14], [2], [15] (cf. the bibliography of [15] for more references).…”

Low regularity classes and entropy numbers

“…Instead of the Pichorides conjecture we shall then use the well known bounds for a martingale analogue, due to Chang, Wilson, and Wolff [2]. This philosophy also applies to the proof of Theorem 1.1; it has been used in other papers, among them [7], [8], [5] (see also references contained in these papers). …”