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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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Additional Information

Werner J. Ricker
Affiliation: Mathematische-Gengrophischen Fakultät, Katholische Universität, Eichstätt- Ingolstadt, D-85072 Eichstätt, Germany

Luis Rodríguez-Piazza
Affiliation: Department Análisis Matemático and IMUS, Facultad de Matemáticas, Universidad de Sevilla, aptdo 1160, E-41080 Sevilla, Spain

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.