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Multidimensional Hausdorff operators on the real Hardy space

Part of: Special classes of linear operators Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  09 April 2009

A. K. Lerner  and
E. Liflyand
A. K. Lerner
Affiliation:
Department of MathematicsBar-Ilan University52900 Ramat-GanIsraelaklerner@netvision.net.il, liflyand@macs.biu.ac.il
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Abstract

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For a wide family of multivariate Hausdorff operators, the boundedness of an operator from this family i s proved on the real Hardy space. By this we extend and strengthen previous results due to Andersen and Móricz.

Keywords

Hausdorff operatorFourier transformreal Hardy space

MSC classification

Secondary: 47B38: Operators on function spaces (general) 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 79 - 86
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Andersen, K. F., ‘Boundedness of Hausdorff operators on Lp(n), H1(n), and BMO(n)’, Acta Sci. Math. (Szeged) 69 (2003), 409–418.Google Scholar
[2]Brown, G. and Móricz, F., ‘The Hausdorff and the quasi Hausdorff operators on the spaces Lp, 1 £ p < ¥’, Math. Inequal. Appl. 3 (2000), 105–115.Google Scholar
[3]Brown, G. and Móricz, F., ‘Multivariate Hausdorff operators on the spaces Lp(R n)’, J. Math. Anal. Appl. 271 (2002), 443–454.CrossRefGoogle Scholar
[4]Coifman, R. R. and Weiss, G., ‘Extensions of Hardy spaces and their use in analysis’, Bull. Amer. Math. Soc. 83 (1977), 569–645.CrossRefGoogle Scholar
[5]Fefferman, C. and Stein, E. M., ‘Hp spaces of several variables’, Ada Math., 129 (1972), 137–193.Google Scholar
[6]Fefferman, R., Some recent developments in Fourier analysis and Hp theory and product domains. II Function spaces and applications Proc. US-Swed. Semin., Lund/Swed., Lect. Notes Math. 1302 (Springer-Verlag, Berlin Heidelberg, 1988) pp. 44–51.Google Scholar
[7]Galanopoulos, P. and Siskakis, A. G., ‘Hausdorff matrices and composition operators’, III. J. Math. 45 (2001), 757–773.Google Scholar
[8]Georgakis, C., ‘The Hausdorff mean of a Fourier-Stieltjes transform’, Proc. Amer. Math. Soc. 116 (1992), 465–471.CrossRefGoogle Scholar
[9]Giang, D. V. and Móricz, F., ‘The Cesaro operator is bounded on the Hardy space H1’, Ada Sci. Math. 61 (1995), 535–544.Google Scholar
[10]Giang, D. V. and Móricz, F., ‘The two dimensional Cesaro operator is bounded on the multi-parameter Hardy space 1(X )’, Acta Sci. Math. 63 (1997), 279–288.Google Scholar
[11]Glazman, I. M. and Lyubich, Yu. I., Finite-Dimensional Linear Analysis: a Systematic Presentation in Problem Form (Nauka, Moscow, 1969 (Russian); English transl.: MIT Press, Cambridge, Massachusets, and London, England, 1974).Google Scholar
[12]Hardy, G. H., Divergent series (Clarendon Press, Oxford, 1949).Google Scholar
[13]Liflyand, E. and Móricz, F., ‘The Hausdorff operator is bounded on the real Hardy space H1()’, Proc. Am. Math. Soc. 128 (2000), 1391–1396.CrossRefGoogle Scholar
[13]Liflyand, E. and Móricz, F., ‘The multi-parameter Hausdorff operator is bounded on the product Hardy space H11(X )’, Analysis 21 (2001), 107–118.CrossRefGoogle Scholar
[15]Miyachi, A., ‘Boundedness of the Cesàro operator in Hardy spaces’, J. Fourier Anal. Appl. 10 (2004), 83–92.CrossRefGoogle Scholar
[16]Móricz, F., ‘Multivariate Hausdorff operators on the spaces H1 (n) and BMO(n)’, Analysis Math. 31 (2005), 31–41.CrossRefGoogle Scholar
[17]Siskakis, A. G., ‘The Cesàro operator is bounded on H1’, Proc. Amer. Math. Soc. 110 (1990), 461–462.Google Scholar
[18]Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals (Princeton Univ. Press, Princeton, NJ, 1993).Google Scholar
[19]Weisz, F., ‘The boundedness of the Hausdorff operator on multi-dimensional Hardy spaces’, Analysis 24 (2004), 183–195.CrossRefGoogle Scholar
[20]Widder, D. V., The Laplace transform (Princeton Univ. Press, Princeton, NJ, 1946).Google Scholar