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Variational estimates for averages and truncated singular integrals along the prime numbers


Authors: Mariusz Mirek, Bartosz Trojan and Pavel Zorin-Kranich
Journal: Trans. Amer. Math. Soc. 369 (2017), 5403-5423
MSC (2010): Primary 37A45; Secondary 42B20, 42B25
DOI: https://doi.org/10.1090/tran/6822
Published electronically: March 30, 2017
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Abstract: We prove, in a unified way, $ r$-variational estimates, $ r>2$, on $ \ell ^{s}(\mathbb{Z})$ spaces, $ s \in (1, \infty )$, for averages and truncated singular integrals along the set of prime numbers. Moreover, we obtain an improved growth rate of the bounds as $ r\to 2$.


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Additional Information

Mariusz Mirek
Affiliation: Mathematical Institute, Universität Bonn, Endenicher Allee 60, D–53115 Bonn, Germany – and – Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Address at time of publication: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Email: mirek@math.ias.edu

Bartosz Trojan
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Address at time of publication: Wydział Matematyki, Politechnika Wrocławska, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: bartosz.trojan@pwr.edu.pl

Pavel Zorin-Kranich
Affiliation: Mathematical Institute, Universität Bonn, Endenicher Allee 60, D–53115 Bonn, Germany
Email: pzorin@math.uni-bonn.de

DOI: https://doi.org/10.1090/tran/6822
Received by editor(s): October 13, 2014
Received by editor(s) in revised form: August 24, 2015
Published electronically: March 30, 2017
Additional Notes: The first and second authors were partially supported by NCN grant DEC–2012/05/D/ST1/ 00053.
The third author was partially supported by the ISF grant 1409/11.
Article copyright: © Copyright 2017 American Mathematical Society