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Pointwise convergence of certain continuous-time double ergodic averages

Part of: Ergodic theory

Published online by Cambridge University Press:  04 May 2021

MICHAEL CHRIST ,
POLONA DURCIK ,
VJEKOSLAV KOVAČ  and
JORIS ROOS
MICHAEL CHRIST
Affiliation:
Department of Mathematics, University of California, Berkeley, CA94720, USA (e-mail: mchrist@berkeley.edu)
POLONA DURCIK
Affiliation:
Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA92866, USA (e-mail: durcik@chapman.edu)
VJEKOSLAV KOVAČ
Affiliation:
Department of Mathematics, Faculty of Science, University of Zagreb, 10000Zagreb, Croatia (e-mail: vjekovac@math.hr)
JORIS ROOS*
Affiliation:
Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, MA01854, USA School of Mathematics, The University of Edinburgh, Edinburgh, EH9 3FD, UK
*
e-mail: joris_roos@uml.edu
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Abstract

We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

Keywords

multiple ergodic averageconvergence almost everywhereCalderón transference principlemultilinear estimate

MSC classification

Primary: 37A30: Ergodic theorems, spectral theory, Markov operators
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 7 , July 2022 , pp. 2270 - 2280
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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