Discrete maximal functions in higher dimensions and applications to ergodic theory

Mariusz Mirek; Bartosz Trojan

doi:10.1353/ajm.2016.0045

Discrete maximal functions in higher dimensions and applications to ergodic theory

Mariusz Mirek, Bartosz Trojan

Research output: Contribution to journal › Article › peer-review

20 Scopus citations

Abstract

We establish a higher dimensional counterpart of Bourgain’s pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates V_r on L^p spaces for all 1 < p <∞ and r > max{p,p/(p−1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

Original language	English (US)
Pages (from-to)	1495-1532
Number of pages	38
Journal	American Journal of Mathematics
Volume	138
Issue number	6
DOIs	https://doi.org/10.1353/ajm.2016.0045
State	Published - Dec 2016
Externally published	Yes

All Science Journal Classification (ASJC) codes

Mathematics(all)

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10.1353/ajm.2016.0045

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title = "Discrete maximal functions in higher dimensions and applications to ergodic theory",

abstract = "We establish a higher dimensional counterpart of Bourgain{\textquoteright}s pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates Vr on Lp spaces for all 1 < p <∞ and r > max{p,p/(p−1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.",

author = "Mariusz Mirek and Bartosz Trojan",

note = "Funding Information: Research supported in part by NCN grant DEC-2012/05/D/ST1/00053. Publisher Copyright: {\textcopyright} 2016 by Johns Hopkins University Press.",

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AB - We establish a higher dimensional counterpart of Bourgain’s pointwise ergodic theorem along an arbitrary integer-valued polynomial mapping. We achieve this by proving variational estimates Vr on Lp spaces for all 1 < p <∞ and r > max{p,p/(p−1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

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Discrete maximal functions in higher dimensions and applications to ergodic theory

Abstract

All Science Journal Classification (ASJC) codes

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