Riesz means of fourier series and integrals: Strong summability at the critical index

Jongchon KIM; Andreas SEEGER

doi:10.1090/tran/7818

Riesz means of fourier series and integrals: Strong summability at the critical index

Jongchon KIM
, Andreas SEEGER

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

3 Citations (Scopus)

Abstract

We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index (d - 1)/2 may fail for functions in the Hardy space h¹(T^d), we prove sharp positive results for strong summability almost everywhere. For functions in L^p(T^d), 1 < p < 2, we consider Riesz means at the critical index d(1/p - 1/2) - 1/2 and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces H^p(R^d) for 0 < p < 1.

Original language	English
Pages (from-to)	2959-2999
Number of pages	41
Journal	Transactions of the American Mathematical Society
Volume	372
Issue number	4
Online published	4 Apr 2019
DOIs	https://doi.org/10.1090/tran/7818
Publication status	Published - 15 Aug 2019
Externally published	Yes

Bibliographical note

Publisher Copyright:
© 2019 American Mathematical Society.

Funding

The first author was supported in part by NSF grants DMS-1500162 and DMS-1638352. The second author was supported in part by NSF grants DMS-1500162 and DMS-1764295. Part of this work was supported by NSF grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. Received by the editors July 29, 2018, and, in revised form, January 21, 2019. 2010 Mathematics Subject Classification. Primary 42B15, 42B25, 42B08. The first author was supported in part by NSF grants DMS-1500162 and DMS-1638352. The second author was supported in part by NSF grants DMS-1500162 and DMS-1764295. Part of this work was supported by NSF grant DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

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title = "Riesz means of fourier series and integrals: Strong summability at the critical index",

abstract = "We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index (d - 1)/2 may fail for functions in the Hardy space h1(Td), we prove sharp positive results for strong summability almost everywhere. For functions in Lp(Td), 1 < p < 2, we consider Riesz means at the critical index d(1/p - 1/2) - 1/2 and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces Hp(Rd) for 0 < p < 1.",

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T2 - Strong summability at the critical index

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AU - SEEGER, Andreas

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N2 - We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index (d - 1)/2 may fail for functions in the Hardy space h1(Td), we prove sharp positive results for strong summability almost everywhere. For functions in Lp(Td), 1 < p < 2, we consider Riesz means at the critical index d(1/p - 1/2) - 1/2 and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces Hp(Rd) for 0 < p < 1.

AB - We consider spherical Riesz means of multiple Fourier series and some generalizations. While almost everywhere convergence of Riesz means at the critical index (d - 1)/2 may fail for functions in the Hardy space h1(Td), we prove sharp positive results for strong summability almost everywhere. For functions in Lp(Td), 1 < p < 2, we consider Riesz means at the critical index d(1/p - 1/2) - 1/2 and prove an almost sharp theorem on strong summability. The results follow via transference from corresponding results for Fourier integrals. We include an endpoint bound on maximal operators associated with generalized Riesz means on Hardy spaces Hp(Rd) for 0 < p < 1.

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Riesz means of fourier series and integrals: Strong summability at the critical index

Abstract

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