Optimal domains and integral representations of Lp (G)-valued convolution operators via measures

Given 1 <= p < infinity, a compact abelian group G and a measure mu is an element of M(G), we investigate the optimal domain of the convolution operator C-mu((p)) : f --> f * mu (as an operator from L-p(G) to itself). This is the largest Kothe function space with order continuous norm into which L-p(G) is embedded and to which C-mu((p)) has a continuous extension, still with values in L-p(G). Of course, the optimal domain depends on p and mu. Whereas C-mu((p)) is compact precisely when mu is an element of M-0(G), this is not always so for the extension of C-mu((p)) to its optimal domain (which is always genuinely larger than L-p(G) whenever mu is an element of M-0(G)). Several characterizations of precisely when the extension is compact are presented. (C) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.