Existence of discrete ergodic singular transforms for admissible processes

Abstract

This article is concerned with the study of the discrete version of generalized ergodic Calderón-Zygmund singular operators. It is shown that such discrete ergodic singular operators for a class of superadditive processes, namely, bounded symmetric admissible processes relative to measure preserving transformations, are weak (1, 1). From this maximal inequality, a.e. existence of the discrete ergodic singular transform is obtained for such superadditive processes. This generalizes the well-known result on the existence of the ergodic Hilbert transform.

Good modulating sequences for the ergodic Hilbert transform

“…for f ∈ L 1 [5,11]. This result has also been extended to various other settings [1,4,6,8,12]. Given a sequence a = {a k } of complex numbers, we will define the modulated ergodic Hilbert transform of f ∈ L p (modulated by a ) as H a f (x) := lim…”

“…The eHt of a symmetric strongly bounded T-admissible process F exists a.e. for all F ⊂ L 1 [6]. There it is also shown that if F = {f n } ⊂ L p is a positive symmetric strongly bounded T -admissible process, then there exists a monotone increasing sequence {v r } ∈ L + p and v r ↑ δ ∈ L p such that f n = T n v |n| for all n ∈ Z, f n ≤ T n δ for all n ∈ Z, and ∥δ∥ p = sup n∈Z ∥f n ∥ p .…”