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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Hausdorff operator is bounded
on the real Hardy space $H^{1} ({\mathbb{R}})$


Authors: Elijah Liflyand and Ferenc Móricz
Journal: Proc. Amer. Math. Soc. 128 (2000), 1391-1396
MSC (1991): Primary 47B38; Secondary 46A30
Published electronically: August 5, 1999
MathSciNet review: 1641140
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Abstract: We prove that the Hausdorff operator generated by a function $\varphi \in L^{1} ({\mathbb{R}}) $ is bounded on the real Hardy space $H^{1} ({\mathbb{R}}) $. The proof is based on the closed graph theorem and on the fact that if a function $f$ in $L^{1} ({\mathbb{R}}) $ is such that its Fourier transform $\widehat {f} (t) $ equals $0$ for $t<0$ (or for $t>0$), then $f\in H^{1} ({\mathbb{R}}) $.


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Additional Information

Elijah Liflyand
Affiliation: Department of Mathematics and Computer Science, Bar-ilan University, 52900 Ramat-gan, Israel
Email: liflyand@macs.biu.ac.il

Ferenc Móricz
Affiliation: Bolyai Institute, University of Szeged, Aradi Vértanúk tere 1, 6720 Szeged, Hungary
Email: moricz@math.u-szeged.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05159-X
PII: S 0002-9939(99)05159-X
Keywords: Fourier transform, Hilbert transform, real Hardy space $H^{1} ({\mathbb{R}}) $, Hausdorff operator, Ces\`{a}ro operator, closed graph theorem
Received by editor(s): June 25, 1998
Published electronically: August 5, 1999
Additional Notes: This research was partially supported by the Minerva Foundation through the Emmy Noether Institute at the Bar-Ilan University and by the Hungarian National Foundation for Scientific Research under Grant T 016 393.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society