On ergodic singular integral operators

Abstract

1. Introduction. In [11] Petersen gave a direct proof of Cotlar's [8] result on the existence a.e. and boundedness of the ergodic Hilbert transform defined by a measure-preserving invertible transformation on a probability space (X, µ). Petersen's proof consists in proving p -inequalities for the maximal discrete Hilbert transform on sequence spaces and then applying Calderón's transference principle ([6], also [7]).In this paper we study a class of operators, called singular series operators (Definition 2.1), which are discrete analogues of singular integral operators on R ([13], [14]). By transference, we then consider the corresponding ergodic operators on L p -spaces of Banach space valued functions on X, for suitable Banach spaces B.In Section 2, the singular series operators are defined as convolution operators on the sequence spaces p , 1 ≤ p < ∞, and we show that the associated maximal operator is bounded on p for 1 < p < ∞ and is of weak type (1, 1). This result can be proved by standard real variable methods using Calderón-Zygmund decomposition. We prove the maximal operator inequalities by transferring these from the corresponding inequalities on R. . For a geometric characterization, we refer to [5].In Section 3, we define the ergodic singular operators and using the transference principle as in [11], we prove the existence a.e. and boundedness of these operators on the spaces L p B (X) consisting of B-valued (strongly)

Existence of discrete ergodic singular transforms for admissible processes

“…Since the Hilbert transform is a special case of discrete ergodic singular transform, and since admissible processes include additive processes, our result generalizes the a.e. existence theorems in [AM,C,Ç 1 ,Ç 2 ,P]. We also observe that if one considers discrete ergodic singular transforms along sequences satisfying the cone condition, then a.e.…”