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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Jongchon Kim and Andreas Seeger
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 42B15, 42B25, 42B08
DOI: https://doi.org/10.1090/tran/7818
Published electronically: April 4, 2019
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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 $ h^1(\mathbb{T}^d)$, we prove sharp positive results for strong summability almost everywhere. For functions in $ L^p(\mathbb{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(\mathbb{R}^d)$ for $ 0<p<1$.


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Additional Information

Jongchon Kim
Affiliation: Institute for Advanced Study, 1 Einstein Drive, Princeton, New Jersey 08540
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada
Email: jkim@math.ubc.ca

Andreas Seeger
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: seeger@math.wisc.edu

DOI: https://doi.org/10.1090/tran/7818
Received by editor(s): July 29, 2018
Received by editor(s) in revised form: January 21, 2019
Published electronically: April 4, 2019
Additional Notes: 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.
Article copyright: © Copyright 2019 American Mathematical Society