Optimal Domains and Integral Representations of Convolution Operators in Lp(G)

Given , a compact abelian group G and a function, we identify the maximal (i.e. optimal) domain of the convolutionoperator (as an operator from L^p(G) to itself). This is thelargest Banach function space (with order continuous norm) into which L^p(G)is embedded and to which has a continuous extension, still with valuesin L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain.Several characterizations of precisely when this is the case are presented.