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Existence of discrete ergodic singular transforms for admissible processes

100%
Çömez D.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 2
335-343
EN
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.
2
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Multiparameter admissible superadditive processes

100%
Çömez D.
Colloquium Mathematicum
|
2005
|
tom 102
|
nr 2
291-305
EN
In this article some properties of Markovian mean ergodic operators are studied. As an application of the tools developed, and using the admissibility feature, a "reduction of order" technique for multiparameter admissible superadditive processes is obtained. This technique is later utilized to obtain a.e. convergence of averages $n^{-2} ∑_{i,j=0}^{n-1} f_{(i,j)}$ as well as their weighted version.
3
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Ergodic averages with generalized weights

63%
Çömez D., Litvinov S.
Studia Mathematica
|
2006
|
tom 173
|
nr 2
103-128
EN
Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified.
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