According to the theory of so-called $A_p$ spaces of Carl Herz the space of convolution operators mapping $L^p(G)$, for $1 \leq p < \infty$ boundedly into itself is characterized by the bounded linear functionals on that space, which is defined as a projective convolution tensor product.

We will describe an approach to this pair of dual spaces in the context of LCA groups, making us of the kernel theorem based on the Banach Gelfand triple of modulation spaces, namely
$M^1,L^2,M^\infty$, or $S_0,L_2,S_0'$ in my usual notation. It makes use of properties of the space $S_0'(GxG)$.