Optimal domains and integral representations of L^p(G)-valued convolution operators via measures

Given 1 ≤ p < ∞, a compact abelian group G and a measure μ ∈ M (G), we investigate the optimal domain of the convolution operator C_μ^(p) : f → f * μ (as an operator from L^p (G) to itself). This is the largest Köthe function space with order continuous norm into which L^p (G) is embedded and to which C_μ^(p) has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and μ. Whereas C_μ^(p) is compact precisely when μ ∈ M ₀(G), this is not always so for the extension of C _μ^(p) to its optimal domain (which is always genuinely larger than L^p(G) whenever μ ∈ M₀ (G)). Several characterizations of precisely when the extension is compact are presented.