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Proceedings of the American Mathematical Society

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The conjugation operator on $A_ q(G)$


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
DOI: https://doi.org/10.1090/S0002-9939-1994-1181167-4
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 \[ {A_q}(G) = \{ f|f \in {L^1}(G),\quad \hat f \in {l_q}(\Gamma )\} \] with the norm ${\left \| f \right \|_{{A_q}}} = {\left \| f \right \|_{{L^1}}} + {\left \| {\hat f} \right \|_{{l_q}}}$. We prove: Let G be a compact, connected abelian group and P any fixed order on $\Gamma$. If $q > 2$ and $\phi$ is a Youngā€™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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Keywords: Conjugation operator, Rudin-Shapiro polynomials
Article copyright: © Copyright 1994 American Mathematical Society