Multipliers on modules over the Fourier algebra

Abstract

Let $G$ be an infinite compact group and $\hat G$ its dual. For $1 \leq p < \infty ,{\mathfrak {L}^p}(\hat G)$ is a module over ${\mathfrak {L}^1}(\hat G) \cong A(G)$, the Fourier algebra of $G$. For $1 \leq p,q < \infty$, let ${\mathfrak {M}_{p,q}} = {\operatorname {Hom} _{A(G)}}({\mathfrak {L}^p}(\hat G),{\mathfrak {L}^q}(\hat G))$. If $G$ is abelian, then ${\mathfrak {M}_{p,p}}$ is the space of ${L^P}(\hat G)$-multipliers. For $1 \leq p < 2$ and $p’$ the conjugate index of $p$, \[ A(G) \cong {\mathfrak {M}_{1,1}} \subset {\mathfrak {M}_{p,p}} = {\mathfrak {M}_{p’,p’}} \subsetneqq {\mathfrak {M}_{2,2}} \cong {L^\infty }(G).\] Further, the space ${\mathfrak {M}_{p,p}}$ is the dual of a space called $\mathcal {A}_p$, a subspace of ${\mathcal {C}_0}(\hat G)$. Using a method of J. F. Price we observe that \[ \cup \{ {\mathfrak {M}_{q,q}}:1 \leq q < p\} \subsetneqq {\mathfrak {M}_{p,p}} \subsetneqq \cap \{ {\mathfrak {M}_{q,q}}:p < q < 2\} \] (where $1 < p < 2$). Finally, ${\mathfrak {M}_{q,p}} = \{ 0\}$ for $1 \leq p < q < \infty$.