The conjugation operator on $A_ q(G)$

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 Young’s function, then the conjugation operator H does not extend to a bounded operator from ${A_q}(G)$ to the Orlicz space ${L^\phi }(G)$.