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A NOTE ON LOCALISED WEIGHTED INEQUALITIES FOR THE EXTENSION OPERATOR

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  01 June 2008

J. A. BARCELÓ ,
J. M. BENNETT  and
A. CARBERY
J. A. BARCELÓ
Affiliation:
ETSI de Caminos, Universidad Politécnica de Madrid, 28040, Madrid, Spain (email: juanantonio.barcelo@upm.es)
J. M. BENNETT*
Affiliation:
School of Mathematics, The Watson Building, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK (email: J.Bennett@bham.ac.uk)
A. CARBERY
Affiliation:
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, JCMB, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK (email: A.Carbery@ed.ac.uk)
*
✉For correspondence; e-mail: J.Bennett@bham.ac.uk
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Abstract

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We prove optimal radially weighted L2-norm inequalities for the Fourier extension operator associated to the unit sphere in ℝn. Such inequalities valid at all scales are well understood. The purpose of this short paper is to establish certain more delicate single-scale versions of these.

Keywords

Fourier extension operatorsweighted inequalities

MSC classification

Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 3 , June 2008 , pp. 289 - 299
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

All authors were supported by the EC project ‘HARP’. The first was also supported by Spanish Grant BFM02206, the second by EPSRC Postdoctoral Fellowship GR/S27009/02 and the third by a Leverhulme Study Abroad Fellowship and EC project ‘Pythagoras’.

References

[1]Barceló, J. A., Bennett, J. M., Carbery, A., Ruiz, A. and Vilela, M. C., ‘Some special solutions to the Schrödinger equation’, Indiana Univ. Math. J. 56(4) (2007), 1581–1593.CrossRefGoogle Scholar
[2]Barceló, J. A., Ruiz, A. and Vega, L., ‘Weighted estimates for the Helmholtz equation and some applications’, J. Funct. Anal. 150(2) (1997), 356–382.CrossRefGoogle Scholar
[3]Bourgain, J., ‘Hausdorff dimension and distance sets’, Israel J. Math. 87 (1994), 193–201.CrossRefGoogle Scholar
[4]Carbery, A. and Soria, F., ‘Pointwise Fourier inversion and localisation in ℝn’, J. Fourier Anal. Appl. 3 (1997), 847–858 (special issue).CrossRefGoogle Scholar
[5]Carbery, A. and Seeger, A., ‘Weighted inequalities for Bochner–Riesz means in the plane’, Q. J. Math. 51 (2000), 155–167.CrossRefGoogle Scholar
[6]Erdoğan, M. B., ‘A note on the Fourier transform of fractal measures’, Math. Res. Lett. 11(2–3) (2004), 299–313.CrossRefGoogle Scholar
[7]Iosevich, A. and Rudnev, M., ‘Distance measures for well-distributed sets’, Discrete Comput. Geom. 38(1) (2007), 61–80.CrossRefGoogle Scholar
[8]Mattila, P., Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, 44 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[9]Mizohata, S., On the Cauchy Problem, Notes and Reports in Mathematics, Science and Engineering, 3 (Academic Press, San Diego, CA, 1985).Google Scholar
[10]Stein, E. M., Harmonic Analysis (Princeton University Press, Princeton, NJ, 1993).Google Scholar
[11]Vargas, A., Personal communication.Google Scholar
[12]Watson, G. N., The Theory of Bessel Functions (Cambridge University Press, Cambridge, 1969).Google Scholar
[13]Wolff, T. H., ‘Decay of circular means of Fourier transforms of measures’, Internat. Math. Research Notices 10 (1999), 547–567.CrossRefGoogle Scholar