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Norm variation of ergodic averages with respect to two commuting transformations

Published online by Cambridge University Press:  17 August 2017

POLONA DURCIK ,
VJEKOSLAV KOVAČ ,
KRISTINA ANA ŠKREB  and
CHRISTOPH THIELE
POLONA DURCIK
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany email durcik@math.uni-bonn.de, thiele@math.uni-bonn.de
VJEKOSLAV KOVAČ
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia email vjekovac@math.hr
KRISTINA ANA ŠKREB
Affiliation:
Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića Miošića 26, 10000 Zagreb, Croatia email kskreb@grad.hr
CHRISTOPH THIELE
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany email durcik@math.uni-bonn.de, thiele@math.uni-bonn.de
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Abstract

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 658 - 688
Copyright
© Cambridge University Press, 2017 

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