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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1181167-4
Keywords: Conjugation operator, Rudin-Shapiro polynomials
Article copyright: © Copyright 1994 American Mathematical Society