Optimal Domains and Integral Representations of Convolution Operators in L ^p(G)

Abstract.

Given $1 \leq p < \infty$, a compact abelian group G and a function $g \in L^1(G)$, we identify the maximal (i.e. optimal) domain of the convolution operator $C^{(p)}_{g} : f \mapsto f * g$ (as an operator from L ^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L ^p(G) is embedded and to which $C^{(p)}_{g}$ has a continuous extension, still with values in L ^p(G). Of course, the optimal domain depends on p and g. Whereas $C^{(p)}_{g}$ is compact, this is not always so for the extension of $C^{(p)}_{g}$ to its optimal domain. Several characterizations of precisely when this is the case are presented.