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The conjugation operator on $ A\sb q(G)$

Authors: Sanjiv Kumar Gupta, Shobha Madan and U. B. Tewari
Journal: Proc. Amer. Math. Soc. 121 (1994), 163-166
MSC: Primary 43A17; Secondary 42A50, 47B38
MathSciNet review: 1181167
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Abstract: Let G be a compact abelian group and $ \Gamma $ its dual. For $ 1 \leq q < \infty $, the space $ {A_q}(G)$ is defined as

$\displaystyle {A_q}(G) = \{ f\vert f \in {L^1}(G),\quad \hat f \in {l_q}(\Gamma )\} $

with the norm $ {\left\Vert f \right\Vert _{{A_q}}} = {\left\Vert f \right\Vert _{{L^1}}} + {\left\Vert {\hat f} \right\Vert _{{l_q}}}$. We prove: Let G be a compact, connected abelian group and P any fixed order on $ \Gamma $. If $ q > 2$ and $ \phi $ is a Young's function, then the conjugation operator H does not extend to a bounded operator from $ {A_q}(G)$ to the Orlicz space $ {L^\phi }(G)$.

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  • [1] Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
  • [2] Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
  • [3] M. A. Krasnosel′skiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. MR 0126722
  • [4] Ronald Larsen, An introduction to the theory of multipliers, Springer-Verlag, New York-Heidelberg, 1971. Die Grundlehren der mathematischen Wissenschaften, Band 175. MR 0435738
  • [5] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • [6] A. Zygmund, Trigonometric series. Vol. I, II, Cambridge University Press, Cambridge-New York-Melbourne, 1977. Reprinting of the 1968 version of the second edition with Volumes I and II bound together. MR 0617944

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Keywords: Conjugation operator, Rudin-Shapiro polynomials
Article copyright: © Copyright 1994 American Mathematical Society

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