The Hausdorff operator is bounded

on the real Hardy space

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 | References | Similar Articles | Additional Information

Abstract: We prove that the Hausdorff operator generated by a function is bounded on the real Hardy space . The proof is based on the closed graph theorem and on the fact that if a function in is such that its Fourier transform equals for (or for ), then .

**1.**R. E. Edwards,*Fourier series: A modern introduction. Vol. I*, Holt, Rinehard and Winston, Inc., New York-Montreal, Que.-London, 1967. MR**0216227****2.**Constantine Georgakis,*The Hausdorff mean of a Fourier-Stieltjes transform*, Proc. Amer. Math. Soc.**116**(1992), no. 2, 465–471. MR**1096210**, 10.1090/S0002-9939-1992-1096210-9**3.**Dang Vu Giang and Ferenc Móricz,*The Cesàro operator is bounded on the Hardy space 𝐻¹*, Acta Sci. Math. (Szeged)**61**(1995), no. 1-4, 535–544. MR**1377382****4.**G. H. Hardy,*Divergent Series*, Oxford, at the Clarendon Press, 1949. MR**0030620****5.**Elias M. Stein and Guido Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR**0304972****6.**E. C. Titchmarsh,*Introduction to the theory of Fourier integrals*, Clarendon, Oxford, 1937.

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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

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