Optimal domains and integral representations of convolution operators in L^P(G)

Given 1 ≤ p <∞, a compact abelian group G and a function g ∈ L1 (G), we identify the maximal (i.e. optimal) domain of the convolution operator C_g ^(p) : f ↦ f * g (as an operator from L^p(G) to itself). This is the largest Danach function space (with order continuous norm) into which L^p(G) is embedded and to which Cg (p) has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas C_g ^(p) is compact, this is not always so for the extension of C_g ^(p) to its optimal domain. Several characterizations of precisely when this is the case are presented.