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Absolutely summing multiplier operators in $ L^p (G)$ for $ p > 2$


Authors: Werner J. Ricker and Luis Rodríguez-Piazza
Journal: Proc. Amer. Math. Soc. 142 (2014), 4305-4313
MSC (2010): Primary 43A15, 47B10; Secondary 43A50, 43A77
Published electronically: August 18, 2014
MathSciNet review: 3266998
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Abstract: Let $ G$ be an infinite compact abelian group. If its dual group $ \Gamma $ contains an element of infinite order, then it is known that, for every $ 4<p<\infty $, there exists a function $ g \in L^p (G) $ whose associated convolution operator $ C_g : f \mapsto f * g $ (on $ L^p (G)$) is absolutely summing but the Fourier series of $ g$ fails to be unconditionally convergent to $ g$ in $ L^p (G)$. It is shown that the restriction on $ \Gamma $ containing an element of infinite order can be removed and also that the range of $ p$ can be extended to arbitrary $ p \in (2, \infty )$.


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  • [1] Bernard Beauzamy and Bernard Maurey, Opérateurs de convolution 𝑟-sommants sur un groupe compact abélien, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A113–A115 (French). MR 0324458
  • [2] B. Beauzamy, Geometrie des espaces de Banach et des opérateurs entre espaces de Banach, PhD Thesis, University of Paris VI, 1976.
  • [3] Andreas Defant and Klaus Floret, Tensor norms and operator ideals, North-Holland Mathematics Studies, vol. 176, North-Holland Publishing Co., Amsterdam, 1993. MR 1209438
  • [4] Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297
  • [5] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
  • [6] 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
  • [7] R. E. Edwards, Fourier series. Vol. 2, 2nd ed., Graduate Texts in Mathematics, vol. 85, Springer-Verlag, New York-Berlin, 1982. A modern introduction. MR 667519
  • [8] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
  • [9] Hermann König, Eigenvalue distribution of compact operators, Operator Theory: Advances and Applications, vol. 16, Birkhäuser Verlag, Basel, 1986. MR 889455
  • [10] Ronald Larsen, An introduction to the theory of multipliers, Springer-Verlag, New York-Heidelberg, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 175. MR 0435738
  • [11] S. Okada, W. J. Ricker, and L. Rodríguez-Piazza, Absolutely summing convolution operators in 𝐿^{𝑝}(𝐺), Proc. Lond. Math. Soc. (3) 102 (2011), no. 5, 843–882. MR 2795726, 10.1112/plms/pdq042
  • [12] S. Okada and W. J. Ricker, Integral extension of multiplier operators in 𝐴(𝐺), Rev. Mat. Complut. 25 (2012), no. 1, 199–219. MR 2876925, 10.1007/s13163-011-0065-8
  • [13] Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
  • [14] P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277

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Additional Information

Werner J. Ricker
Affiliation: Mathematische-Gengrophischen Fakultät, Katholische Universität, Eichstätt- Ingolstadt, D-85072 Eichstätt, Germany
Email: werner.ricker@ku-eichstaett.de

Luis Rodríguez-Piazza
Affiliation: Department Análisis Matemático and IMUS, Facultad de Matemáticas, Universidad de Sevilla, aptdo 1160, E-41080 Sevilla, Spain
Email: piazza@us.es

DOI: https://doi.org/10.1090/S0002-9939-2014-12179-4
Keywords: Absolutely summing operator, $p$-multiplier operator, Fourier series
Received by editor(s): January 31, 2013
Published electronically: August 18, 2014
Additional Notes: The second author was partially supported by the Spanish government and European Union (FEDER), project MTM 2012-30748
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.