Radial functions and invariant convolution operators

Abstract:

For $1 < p < 2$ and $n > 1$, let ${A_p}({{\mathbf {R}}^n})$ denote the Figà-Talamanca-Herz algebra, consisting of functions of the form $( \ast )$ \[ \sum \limits _{k = 0}^\infty {{f_k} \ast {g_k}} \] with $\sum \nolimits _k {||{f_k}|{|_p}\cdot ||{g_k}|{|_{p\prime }} < \infty }$. We show that if $2n/(n + 1) < p < 2$, then the subalgebra of radial functions in ${A_p}({{\mathbf {R}}^n})$ is strictly larger than the subspace of functions with expansions $( \ast )$ subject to the additional condition that ${f_k}$ and ${g_k}$ are radial for all $k$. This is a partial answer to a question of Eymard and is a consequence of results of Herz and Fefferman. We arrive at the statement above after examining a more abstract situation. Namely, we fix $G \in [FIA]_{B}^{ - }$ and consider $^B{A_p}(G)$ the subalgebra of $B$-invariant elements of ${A_p}(G)$. In particular, we show that the dual of $^B{A_p}(G)$ is equal to the space of bounded, right-translation invariant operators on ${L^{p}}(G)$ which commute with the action of $B$.