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Restriction and extension of Fourier multipliers between weighted $ L^p$ spaces on $ \mathbb{R}^n$ and $ \mathbb{T}^n$


Authors: Kenneth F. Andersen and Parasar Mohanty
Journal: Proc. Amer. Math. Soc. 137 (2009), 1689-1697
MSC (2000): Primary 42B15
DOI: https://doi.org/10.1090/S0002-9939-08-09774-8
Published electronically: December 29, 2008
MathSciNet review: 2470827
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Abstract: Weighted analogues of de Leeuw's restriction theorem for Fourier multipliers on $ L^p(\mathbb{R}^n)$ are obtained. Weighted analogues of related extension theorems for multipliers on $ L^p(\mathbb{T})$ are also considered.


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  • 1. N. Asmar, E. Berkson, and J. Bourgain, Restrictions from $ \mathbb{R}^N$ to $ \mathbb{Z}^N$ of weak type (1,1) multipliers. Studia Math. 108(1994), 291-299. MR 95b:42015
  • 2. N. Asmar, E. Berkson, and T.A. Gillespie, Generalized de Leeuw's theorems and extension theorems for weak multipliers. Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math., 175, 41-67, Dekker, New York, 1996. MR 96j:43002
  • 3. E. Berkson and T.A. Gillespie, On restrictions of multipliers in weighted settings. Indiana Univ. Math. J. 52(2003), 927-961. MR 2005i:43006
  • 4. E. Berkson, M. Paluszyňski, and G. Weiss, Transference couples and their applications to convolution operators and maximal operators. Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math., 175, 69-84, Dekker, New York, 1996. MR 96j:43005
  • 5. D. Chen and D. Fan, Multiplier transformations on $ H^p$ spaces. Studia Math. 131(1998), 189-204. MR 99e:42032
  • 6. K. de Leeuw, On $ L^p$ multipliers. Ann. of Math. (2) 81(1965), 364-370. MR 30:5127
  • 7. R. E. Edwards and G. I. Gaudry, Littlewood-Paley and multiplier theory, Springer-Verlag, Berlin-New York, 1977. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90. MR 0618663
  • 8. A. Figá-Talamanca and G.I. Gaudry, Multipliers of $ L^p$ which vanish at infinity. J. Functional Analysis 7(1971), 475-486. MR 43:2429
  • 9. M. Jodeit, Restrictions and extensions of Fourier multipliers. Studia Math. 34(1970), 215-226. MR 41:7376
  • 10. P. Mohanty and S. Madan, Summability kernels for $ L^p$ multipliers. J. Fourier Anal. Appl. 9(2003), 127-140. MR 2003m:42019
  • 11. B. Muckenhoupt, R. Wheeden, and W. Young, $ L^2$ multipliers with power weights. Adv. in Math. 49(1983), 170-216. MR 85d:42010
  • 12. E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1971. MR 46:4102

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Additional Information

Kenneth F. Andersen
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada
Email: Ken.Andersen@ualberta.ca

Parasar Mohanty
Affiliation: Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, U.P. 208016, India
Email: parasar@iitk.ac.in

DOI: https://doi.org/10.1090/S0002-9939-08-09774-8
Received by editor(s): March 3, 2008
Published electronically: December 29, 2008
Additional Notes: This research was supported in part by the University of Alberta Faculty of Science Research Allowance
Communicated by: Hart F. Smith
Article copyright: © Copyright 2008 American Mathematical Society

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