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

DOI:
https://doi.org/10.1090/S0002-9939-99-05159-X

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*, Holt, Rinehart and Winston, New York, 1967. MR**35:7062****2.**C. Georgakis,*The Hausdorff mean of a Fourier-Stieltjes transform*, Proc. Amer. Math. Soc.**116**(1992), 465-471. MR**92m:42009****3.**D. V. Giang and F. Móricz,*The Cesàro operator is bounded on the Hardy space*, Acta Sci. Math. (Szeged)**61**(1995), 535-544. MR**96m:47051****4.**G. H. Hardy,*Divergent series*, Clarendon, Oxford, 1949. MR**11:25a****5.**E. M. Stein and G. Weiss,*Introduction to Fourier analysis on Euclidean spaces*, Princeton Univ. Press, New Jersey, 1971. MR**46:4102****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:
https://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